Sa = Sb --> a = b

where S is the successor function. How does one establish from the axioms that S is, in fact, a function, that is the converse

a = b --> Sa = Sb?

Probably a very simple matter, but I would appreciate any help in clarifying. Many thanks in advance,

Agapito]]>

I want to construct a circuit that implements the following truth table:

\begin{equation*}

\begin{array}{c|c|c|c|c}

P(Input) & Q(Input) & K(Output) & A(Output) \\

\hline

1 & 1 & 1 & 0 \\

1 & 0 & 0 & 1 \\

0 & 1 & 0 & 1 \\

0 & 0 & 0 & 0

\end{array}

\end{equation*}

In addition, I want to use the above circuit to construct a full adder, i.e. a circuit that adds three 1-digit binary numbers.

Could you help me to construct the wanted circuit that implements the...

Construct circuit that implements truth table]]>

you go to the warehouse and randomly select 80 clocks.

1. How many of the 80 clocks do you expect to be defective?

2.What is the probability that exactly 6 of the clocks are defective?

3. What is the probability that at least one of the clocks (out of 80) is defective? use the complement.]]>

I want to calculate the truth tables of the following propositions:

$$(p \land q) \lor (\lnot p \land q) \to q \\ p \land \lnot q \to r$$

I have done the following:

\begin{equation*}

\begin{array}{c|c|c|c|c}

p & q & p \land q & \lnot p \land q & (p \land q) \lor (\lnot p \land q) \to q \\

\hline

1 & 1 & 1 & 0 & 1 \\

1 & 0 & 0 & 0 & 1 \\

0 & 0 & 0 & 0 & 1 \\

0 & 1 & 0 & 1 & 1

\end{array}

\end{equation*}

and

\begin{equation*}

\begin{array}{c|c|c|c}

p & q & \lnot q...

Truth tables]]>

We suppose that the propositions $p,q$ are propositions such that the proposition $p \to q$ is false.

Find the truth values for each of the following propositions:

- $\sim q \to p$
- $p \land q$
- $q \to p$

I have thought the following:

Since the proposition $p \to q$ is false, either $p$ is true and $q$ is false, either $q$ is true and $p$ is false.

Thus, we have the following truth table:

$\begin{matrix}

p & q & \sim q & \sim q \to p & p \land q &...

Find truth value of propositions]]>

The result of substituting the quotation of “The result of substituting the quotation of x for ‘x’ in x has property P.” for ‘x’ in “The result of substituting the quotation of x for ‘x’ in x has property P.” has property P.

Supposedly this ends up in a sentence that “says of itself it has Property P” in the sense that it says that a sentence satisfying a certain description has property P, and the sentence itself is the only one and...

Godelian self-reference]]>

I am havinga bit of challenge with the following question.

What seems confusing to me is th relationship between hypothesis 1 and 2.

I will appreciate all help.

]]>

I'm currently confuse in this one help will be very much needed]]>

I request you to solve 2 questions ( q-3 and q-5 ) of symbolic logic ( Strenthened method of conditional proof ).

These questions are taken from I.M.Copi's 'symbolic logic' ( edition -5, sec. 3.8, pg- 61 )

File is being attached.

thank you

yours truly

Deep Kumar Trivedi]]>

I have some tasks that I want someone could help for me to solve.

Step-by-step solutions would be really good, to really know how these tasks can be done (main purpose is to gain more knowledge in these things):

Letters $C$, $B$ and $D$ mark these facts: $C$ = "R is a student"; $B$ = "G is a sstudent"; $D$ = "P is a...

Please help with bool algebra and logic things!]]>

In honor of Pi Day I'm going to be explaining the very beginning of set theory (which I consider the beginning of university math) live on Twitch in about two hours (1 PM GMT).

For those who do not know Twitch, it's a completely free streaming platform - you can come in and watch without registering or anything.

Starting university math can often be confusing so I'm hoping to be of some help to people with this.

You can find the stream here...

Introduction to Set Theory Stream]]>

$|N(v) \cap S| \geq t$ for all $v \in V (G)$.

Suppose that r, t, n are positive integers such that $r > 2t$ and $t \geq \frac{14}{3}\cdot ln(2n)$, and let $G$ be an r-regular n-vertex graph. Choose a random vertex set $S$ by independently adding each vertex to $S$ with probability p, where $p =\frac{2t}{r}$. Prove that $P[S$ is totally t-dominating$] > \frac{1}{2}$.

I honestly am...

Probability of Having a Totally Dominating Set (Probabilistic Methods)]]>

Suppose that the function $p : N \rightarrow [0, 1]$ satisfies $p >> n^{-1}ln(n)$ (i.e. $n^{-1}ln(n) = o(p)$).

I am not getting to where I think I should...

Minimum Degree of a Random Graph (Probabilistic Method)]]>

Apparently this is the "dual version" of Turan's Theorem. How does this theorem imply Turan's?

That $ex(n, K_{p+1}) = |E(T(n,p))|$

Where:

- $T'(n,q)$ : $q$ disjoint cliques with size as equal as possible

- $\alpha (G)$ : independence number of $G$

- $ex(n, K_{p+1})$ : max number of edges in an n-vertex...

Turans Theorem (Dual?)]]>

I am looking at the following:

There are the terms reflexive, symmetric, antisymmetric and transitive.

Give for each combination of the properties (if possible) a set $M$ and a relation $R$ on $M$, such that $R$ satisfies these properties.

What is meant exactly? Every possible combination? So do we have to give a set and a relation that satisfies the below properties?

- reflexive, symmetric
- reflexive, antisymmetric
- reflexive, transitive
- symmetric...

Give a set and a relation that satisfies the properties]]>

I look at the problem with hilbert's hotel and the busses. First one bus with infinitely many guests, then four busses with infinitely many guests and then infinitely many busses with infinitely many guests.

At each case we move all the guests that are already in the hotel to the odd room numbers.

So the even room numbers are free for the new guests.

In the first with one bus we use the formula $n = 2(i-1)+2$ where $i\in \mathbb{N}\setminus\{0\}$ is the number of place in the...

Hilbert's hotel and busses : Injectivity - Surjectivity]]>

1) P ⊢ P

2) P → Q, Q→R ⊢ P → R

3) P → Q, Q→R, ¬R ⊢ ¬P

4) Q→R ⊢ (PvQ) → (PvR)

5) P →Q ⊢ (P&R) → (Q&R)]]>

The answer is false, but I don't know how to justify it. I would appreciate any help.]]>

Let $M:=\{1, 2, \ldots, 10\}$ and $\mathcal{P}:=\{\{1,3,4\}, \{2,8\}, \{7\}, \{5, 6, 9, 10\}\}$.

For $x \in M$ let $[x]$ be the unique set of $\mathcal{P}$ that contains $x$.

We define the relation on $M$ as $x\sim y:\iff [x]=[y]$.

Show that $\sim$ is an equivalence relation.

For that we have to show that the relation is reflexive, symmetric and transitive.

- Reflexivity:

Let $x \in M$. Then it holds, trivially, that $[x]=[x]$. Therefore $x\sim x$. So $\sim$...

Show that ~ is an equivalence relation]]>

I want to determine the following sets:

- $\displaystyle{\bigcap_{1\leq n\in \mathbb{N}}\left (-\frac{1}{n},\frac{1}{n}\right )}$

- $\displaystyle{\bigcup_{n\in \mathbb{N}}\left (-n,n\right )}$

- $\displaystyle{\bigcap_{n\in \mathbb{N}}\left (n, 10n^2+50\right )}$

I have done the following:

- Let $\displaystyle{x\in \bigcap_{1\leq n\in \mathbb{N}}\left (-\frac{1}{n},\frac{1}{n}\right )}$. This means\begin{align*}&\forall n\in \mathbb{N}...

Intesection of intervals]]>

set of consistent $\mathcal L$-formulas, then $\Phi$ is satisfiable.

How is that constructed? There are a large number of Lemmas working from Machover's text Set theory, Logic and Their Limitations but I'm having trouble with which are most relevant and how it comes together.]]>

(i) There is at least one natural number.

(ii) For each natural number there is a distinct number which is its successor, i.e., for each number $x$ there is a distinct number $y$ such that $y$ stands in the

successor relation to $x$.

(iii) No two natural numbers have the same successor.

(iv) There is a natural number, namely 0, that is not the successor of any number.

Bearing these...

First-order logic formula satisfiable only if the domain of the valuation is infinite?]]>

- Let $M:=\{7,4,0,3\}$. Determine $2^M$.
- Prove or disprove $2^{A\times B}=\{A'\times B'\mid A'\subseteq A, B'\subseteq B\}$.
- Let $a\neq b\in \mathbb{R}$ and $M:=2^{\{a,b\}}$. Determine $2^M$.
- Is there a set $M$, such that $2^M=\emptyset$ ?

First of all how is $2^M$ defined? Is this the powerset? ]]>

Let $A,B$ be sets, such that $A\times B=B\times A$. I want to show that one of the following statements hold:

- $A=B$
- $\emptyset \in \{A,B\}$

I have done the following:

Let $A$ and $B$ be non-empty set.

Let $a\in A$. For each $x\in B$ we have that $(a,x)\in A\times B$. Since $A\times B=B\times A$, it follows that $(a,x)\in B\times A$. So $a\in B$.

That means that $A\subseteq B$.

Let $b\in B$. For each $y\in A$ we have that $(y,b)\in A\times B$...

Sets so that the cartesian product is commutative]]>

1. $\sigma \vDash \alpha \rightarrow \forall x\alpha$ where $x$ does not occur in a free $\alpha$

2. $\sigma \vDash s_1 = t_1 \rightarrow ... \rightarrow s_n = t_n \rightarrow fs_1...s_n=ft_1...t_n$

3. $\sigma \vDash \forall x \alpha \rightarrow \alpha(x/t)$ (appealing to the fact that generally $\alpha(x/t)^\sigma = {\alpha}^{\sigma(x/t^\sigma )})$]]>

A square with the side length $2^n$ length units (LU) is divided in sub-squares with the side length $1$. One of the sub-squares in the corners has been removed. All other sub-squares should now be covered completely and without overlapping with L-stones. An L-stone consists of three sub-squares that together form an L.

I want to draw the problem for the first three cases described above ($1 \leq n \leq 3$).

Then I want to show the following using induction:

For all $n \in N$...

Induction: Each square can be covered by L-stones]]>

Hey So I am trying to prove this.

I tried using linear combinations and not sure how that would help. I am just not familiar with combinatorics and wondering if anyone would enlighten me.]]>

1. $|S|<O(G−S)$

2. Each vertex in S is adjacent to vertices in at least three odd components of $G−S$.

(where $O(G−S)$ is the number of odd components of $G−S$)

Number 1. is the contrapositive of Tutte's 1-Factor Theorem. So no problems there.

Should number 2. also be proved using the contrapositive? Meaning to show...

A Subset of a Graph with No Perfect Matching]]>

1. T(n) = 3T([n/3])+n

2. T(n) = T([n/4])+T([n/3])+n

3. T(n) = 2T([n/4])+√n]]>

I need some help with this task. My theory book only shows examples of how to solve sequences in the form :

𝑎𝑘 = A * 𝑎(𝑘−1) − B * 𝑎(𝑘−2).

But I've no idea how to solve this task because of the alternating term. I've included the Answer (called "Svar") to the task.]]>

I am looking the follwong exercise:

Using the method of Quine-McCluskey, determine the prime implicants for the following switching function and find a disjunctive minimal form. If available, also specify all other disjoint minimal forms.

The switching function is:

\begin{align*}f(x_1, x_2, x_3, x_4, x_5)&=\bar{x}_1\bar{x}_2\bar{x}_3\bar{x}_4\bar{x}_5\lor \bar{x}_1x_2\bar{x}_3\bar{x}_4\bar{x}_5\lor \bar{x}_1\bar{x}_2\bar{x}_3\bar{x}_4x_5\lor \bar{x}_1x_2\bar{x}_3x_4\bar{x}_5...

Prime implicants and disjunctive minimal form]]>

I am having some difficulties on exercise 2e from Topology 2nd ed by J. Munkres . Here are the directions:

determine which of the following states are true for all sets [FONT=MathJax_Math-italic]A[/FONT], [FONT=MathJax_Math-italic]B[/FONT], [FONT=MathJax_Math-italic]C[/FONT], and...

Topology Munkres Chapter 1 exercise 2 e- Set theory]]>

I am having some difficulties on exercise 2b and 2c from Topology 2nd ed by J. Munkres . Here are the directions:

determine which of the following states are true for all sets $A$, $B$, $C$, and $D$. If a double implication fails, determine whether one or the other one of the possible implication holds. If an equality fails, determine whether the statement becomes true if the "equal" symbol is replaced by one or the other of the inclusion symbols $\subset$ or $\supset$...

Topology Munkres Chapter 1 exercise 2 b and c- Set theory equivalent statements]]>

2) $\cap_{i=1}^{\infty} A_{i}= A_{1}\setminus \cup_{i=1}^{\infty}(A_{1}\setminus A_{i})$]]>

In my question, Gödel numbers are used to encode wffs as follows:

Syntactically (by formalism without semantics) there is set A (the set which is postulated to be infinite), such that the empty set is a member of A, and such that whenever any

So...

Do Gödel numbers can be used to derermine the usefulness of an infinite set as a complete whole?]]>

I am trying to prove a simple thing, that if AxA = BxB then A=B.

The intuition is clear to me. If a pair (x,y) belongs to AxA it means that x is in A and y is in A. If a pair (x,y) belongs to BxB it means that x is in B and y is in B. If the sets of all pairs are equal, it means that every x in A is also in B and vice versa.

How do I prove it formally ?

Thank you !]]>

Are the following questions decidable? Give an appropriate algorithm or show that the problem is undecidable using Rice's theorem.

- Does an arbitrary Turing machine halt with all inputs less than or equal to 1000 within the first 1000 steps?

- Does an arbitrary algorithm with the appropriate input give a number, that interpreted as a string, it is palindrome?

- Does an arbitrary algorithm, that does not halt for at least one input, give any natural number as an...

Are the questions decidable?]]>

I have two small questions regarding operations on sets.

(1) Prove that \[A\subseteq B\subseteq C\] if and only if \[A\cup B=B\cap C\].

(2) What can you say about sets A and B if \[A\B = B\] ?

In the case of (1), I have used a Venn diagram and I understand why it is true, but struggle to prove it.

In the case of (2) I think it means that B is an empty set , am I correct ?

Thank you !]]>

Are the following instances of post correspondence problem solvable? Find, if possible, various non-periodic solutions or prove the insolubility.

1. $\alpha = (bb, ab, aab)$, $\beta = (abb, a, bba)$

2. $\alpha = (bab, baa, a, aabbb)$, $\beta = (b, a, aba, ababbb)$

For the first one I tried various sequences but I didnt find a common sequence so I suppose that this one is not solvable. Is that correct?

To prove that, we suppose that the problem is solvable.

There exists a...

Are the instances of PCP solvable?]]>

The code words of a linear code $C$ have the length $n=5$.

Writing the code words into a matrix to get the linear independent ones, we get the following:

\begin{equation*}\begin{pmatrix}0 & 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 1 & 0 & 1 & 1 & 1 \\ 1 & 1 & 0 & 1 & 0\end{pmatrix}\rightarrow \ldots \rightarrow \begin{pmatrix}1 & 1 & 0 & 1 & 0 \\ 0 & 1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0\end{pmatrix}\end{equation*}

So the dimension of $C$ is $m=2$.

Wir have...

Linear Codes : Error detection]]>

By drawing a series of connected straight lines from dot to dot, it is possible to divide the square (which effectively has side length $3$) into two parts of equal area. One way of doing this is as shown:

Other examples of ways of doing this are: ; ; and...

Number of ways to divide a square grid in half]]>

(∀v Fv -> p) <=> (∃u Fu -> p)

Where variable v occurs free in Fv at all and only those places that u occurs free in Fu, and p is a proposition containing no free occurences of variable v.

Can someone please offer a proof of such equivalence. Many thanks. am

I'm working on a discrete mathematics for computing paper and am stuck on what a symbol is trying to convey. Sorry if this seems like a stupid question (I feel stupid for not being able to work it out myself), I've just started this subject and am still getting used to it.

My question is what does the Xc mean in this algorithm (picture attached)? I understand that X1, X2 etc are the different variables, but does the Xc have something to do with the variable C := 1? If so, what does...

Discrete maths problem - tracing an algorithm]]>

I am looking the following:

Determine a disjunctive normal form for the expression $$x_1\left (\left (x_2x_3\lor \bar{x}_2x_4\right )\lor x_2\lor x_3x_4\right )\lor x_2\bar{x}_4\bar{\left (\bar{x}_1x_3\right )}\lor x_1x_3\bar{x}_4$$ and minimize it using a Karnaugh Diagram.

The above expression is equivalent to $$ x_1x_2x_3\lor x_1\bar{x}_2x_4\lor x_1x_2\bar{x}_3x_4\lor x_2\bar{x}_4x_1\lor x_2\bar{x}_4\bar{x}_3\lor x_1x_3\bar{x}_4$$

Does each term has t contain all the...

Karnaugh Diagram]]>

I am looking at the following:

Use the Quine-McCluskey method to determine the respective prime implicants for the following boolean functions and find a disjunctive minimal form. If available, also give all others disjunctive minimal forms.

\begin{equation*}f(x_1, x_2, x_3, x_4)=\bar{x}_1x_2(\bar{x}_3\lor x_3\bar{x}_4)\lor x_1(x_3\bar{x}_4\lor x_2)\lor (x_1\bar{x}_2\lor x_2)x_3\lor \bar{x}_1x_2(\bar{x}_3x_4\lor \bar{x}_3\bar{x}_4\lor x_3x_4)\end{equation*}

First, we open the...

Quine-McCluskey method: Prime implicants - disjunctive minimal forms]]>

2. (I v X) ≡ (R ⊃ P) P

3. R ⊃ (I ⊃ (N ^ V)) P

4. R A

5. Reiteration of 3

6. I ⊃ (N ^ V) 3,4 ⊃E

7. I A

8. I v X 7, vI

9. Reiteration of 2

10. R ⊃ P 8,9 ≡E

11. R 4R

12. P 10,11 ⊃E

13. I ⊃ P 7-12...

Stuck on Derivation (Natural Deduction)]]>

prove the following sequent:

1. $(\exists x) Fx \to (\forall x) Gx \vdash (\exists x)(Fx \to (\forall x)Gx)$

2. $(\forall x)(Fx \to (\forall y)\neg Fy) \vdash \neg(\exists x)Fx$

3. $(\exists x)Fx, (\forall x)(Fx \; à \; Gx) \vdash (\exists x)G$]]>

We have the following equalities:

\begin{align*}\left (\overline{y}\land z\lor x\land \left (\overline{z}\lor y\right )\right )\land \left (\overline{x}\lor \overline{y}\right )& \overset{(1)}{=}\left (\overline{y}\land z\lor x\land \overline{z}\lor x\land y\right )\land \left (\overline{x}\lor \overline{y}\right ) \\ & \overset{(2)}{=}\left (\overline{y}\land z\lor x\land \overline{z}\right )\land \left (\overline{x}\lor \overline{y}\right ) \\ & \overset{(3)}{=}\left...

Two-element Boolean algebra: How are the equalities derived?]]>

For example, logical disjunction (OR) and set-theoretic UNION are isomorphic in this sense:

0 OR 0 = 0.

{0} UNION {0} = {0}.

Similarly, logical AND & set theoretic INTERSECTION are isomorphs:

0 AND 0 = 0.

{0} INTERSECTION {0} = {0}.

Anyway,, has anybody ever heard of this? I'm trying to figure out how the isomorphism comprises arithmetic also, whereby...

isomorphism of logic, arithmetic, and set theory]]>

Given the domain as:

D = {a,b}; ~Ba & Bb & Laa & ~Lab & Lba & ~Lbb

Why is the interpretation false? (∀x)[Bx ⊃ (Lxx ⊃ Lxa)]

I am having trouble understanding why that is the case because (Lxx ⊃ Lxa) evaluates to true in any case as long as Lxa is true in all cases, so the overall interpration should be true in all cases.

The false case that is given is: Ba ⊃ (Laa ⊃ Laa), but isn't this case true as well?]]>

random order. Show that no matter what this order is, there must be three successive

integers whose sum is at least 45.]]>

$\newcommand{\R}{\mathbb{R}}$

I am struck in writing the equivalence classes. And the problem is this:

Let ${\R}^{2}= \R \times \R$. Consider the relation $\sim$ on ${\R}^{2}$ that is given by $({x}_{1},{y}_{1}) \sim ({x}_{2},{y}_{2})$ whenever ${y}_{1}-{{x}_{1}}^{3}={y}_{2}-{{x}_{2}}^{3}$. Prove that $\sim$ is an equivalence relation. What are the equivalence classes?

I have proved that relation is an equivalence relation.

Here is my attempt...

Equivalence Class]]>

I am struggling now for determining if the following statements are true or false. If the statement is true, then prove it. If not, make a counterexample.

Here are the statements:

- $A \subset B$ and $A \subset C$ if and only if $A \subset (B\cup C)$.
- $A \subset B$ or $A \subset C$ if and only if $A \subset (B\cap C)$.

My attemption:

Let A={1,2,3}, B={1,2,4}, and C={3}.

- I believe this is true. $A\subset B$ and $A\subset C$. Thus $A \subset...

Set Theory]]>

(x = y) ----> [(y=x) <---> (y=y)] ?

All help appreciated, am

This is my attempt at the question.

$1. P \land (R\implies Q) $ Premise

$2. ( P \land Q ) \implies \lnot S) $ Premise

$3. ( P \land S) \implies R) $ Premise

$4. \lnot S $ Conclusion

$5. P \land Q$ $ \beta 2,4$

$6. P $ $ \alpha 5$

$7. Q$ $...

KE Deduction]]>

Any other inputs kind help. It will improve my knowledge way of my thinking.

suppose (V,O_1,O_2) and (V,a_1,a_2) are two different vector spaces on the same set V can they both have same bases. (If not kind help with a proof or link of it to improve my knowledge.)

Here O_1,O_2,a_1,a_2 are operations on V.]]>

The question is attached.

I know that " $\implies $ " (implies) has precedence from right to left. But because " l- " appears after

P$\implies ($Q $\implies$ R ), in my truth table do I evaluate:

(P$\implies ($Q $\implies$ R ) ) $\implies$ ((P$\implies$Q ) $\implies$ R ) )

or

P$\implies ($Q $\implies$ R ) $\implies$ (P$\implies$Q ) $\implies$ R )

Thank you for any help. ]]>

Am new to this forum, but I have looked around here for some time now, since am studying a course of logic in the context of computer science. I have a very important exam in a few days, and while I thought I got it, I got shocked when I was looking on previous graded exams to see what I could work on in the final days. And when I came to the part about resolution method and counterexamples for predicate logic on one exam I realised that either there is something I don't understand, or...

Resolution method and counterexample]]>

\text{ Let }

π ∈ S_{n} \text{ and z } \text{ the number of disjunctive Cycles of π. Here will be counted 1 - Cycle }. (a) \text{ Prove that } sgn (π) = (-1)^{n-z}.

(b) \text{ Prove that subset } A_{n}= \{π∈Sn∣sgn(π)=1\} ⊆ S_{n}\text{ is subgroup of } S_{n}.

(c)

\text{ Find number of elements } |A_{n}| \text{ of a subgroup } A_{n} \text{ from (b) }

$$]]>

\text{ Prove that always exist } i,j∈ \underline{n} \text{ with } i≤j \text{ so }

\sum\limits_{k=i}^{\\j} a_{k} \text{ divisible by n} .$$]]>

Q2: Let S = {1,2,5,6 }

Define a relation R on S of at least four order pairs, as (a,b) R iff a*b is even (i.e. a multiply by b is even)

Q3: Let S = {x, y, z } and R is a relation defined on S such that...

proper sub set]]>

Define a relation R on S of at least four order pairs, as (a,b) R iff a*b is even (i.e. a multiply by b is even)]]>

Q: Let S = {1,2,5,6 }

Define a relation R on S of at least four order pairs, as (a,b) R iff a*b is even (i.e. a multiply by b is even)]]>

Generally, I have some difficulty with the concept of recursion, as well as with the recursion in programming unfortunately.

I have some question to solve, and maybe you can guide me:

Find a recursive formula and a terminal condition for the number of words with the length 'n' that can be written by $A, B, C$, such that these combinations won't be shown: $AB, AC, BA, BC$.

Now, I know that if we start from...

Find a recursive formula for the number of combinations]]>

R={(y,y),(x,z),(z,x),(x,x),(z,z),(x,y),(y,x)}

Show that R is reflexive and symmetric as well.]]>

The series $a_n$ is defined by a recursive formula $a_n = a_{n-1} + a_{n-3}$ and its base case is $a_1 = 1 \ a_2 = 2 \ a_3 = 3$.

Prove that every natural number can be written as a sum (of one or more) of different elements of the series $a_n$.

Now, I know that is correct intuitively but I don't know how to prove that.

Generally, I have some problem of understanding the concept of recursion.

Thanks.]]>

We started to study all this subject of combinatorics integrated with the subject of functions.

1. I don't actually understand how to integrate between combinatorics and function, those functions which represent our possibilities and etc...

And why at all we need to represent our combinatoric problem with an answer integrating functions?

2. Secondly, we get this formula to calculate some sorts of possibilities:

$\frac {n!} {(n-k)!}$

It's not clear to me why is this really correct...

Some fundamental question in combinatorics]]>

I find it difficult to understand the logic and the appropriate usage of the formula:

$\dbinom{N+K-1}{N}$

I don't really understand what's posed behind the scenes of that one.

So I have some example for an exercise which requires the usage of this formula, but I know only to substitute robotically numbers without any understanding.

* How many ways we can insert 16 indistinguishable balls to 4 drawers such that in every drawer we have at least 3 balls?

So I know we should insert at...

Stars and bars selection]]>

* How many ways we can insert 16 similar balls to 4 drawers such that in every drawer we have at least 3 balls?

So I know we have to insert at first 3 balls to every drawer and the remainder is 4 balls.

But how can I calculate the amount of possibilities to insert 4 balls to 4 drawers without any limit?

Thanks.]]>

(a) A sorting algorithm takes one operation to sort an array with one item in it.

Increasing the number of items in the array from n to n + 1 requires at most an

additional 2n + 1 operations. Prove by induction that the number of operations

required to sort an array with n > 0 items requires at most n^2 operations.

(b) Show by induction that if n ≥ 1, then 7n − 1 is a multiple of 6.

So I'm confused to what the question even is and how...

Proof by induction - Really confused]]>

We use the predicates O and B, with domain the integers. O(n) is true if n is odd, and

B(n) is true if n if big, which here means that n > 100.

(a) Express ∀m, n ∈ Z|O(m) ∧ B(n) ⇒ B(n − m) in conversational English.

(b) Find a counter-example to this statement.

Now i take it two ways;

Just expressing the imply part of the statement so,

The difference between n and m is big.

Or is it deeper than that like;

For every m, n is an...

Predicates and Quanitifiers - Can't understand Question]]>

If $A, B$ are sets and if $A \subseteq B$, prove that $|A| \le |B|$.

Proof:

(a) By definition of subset or equal, if $x \in A$ then $x \in B$. However the converse statement if $x \in B$ then $x \in A$ is not always well defined.

(b) Therefore the identity function $f: A \rightarrow B$ defined by $f(x) = x$ is only an injection. Hence by theorem on the textbook, $|A| \le |B|$.

Thank you for...

Inequality of Cardinality of Sets]]>

I am working on a set equivalent (the textbook refers as "equinumerous" denoted by ~) as follows:

If $S$ and $T$ are sets, prove that if $(S\backslash T) \sim (T\backslash S)$, then $S \sim T$.

Here is my own proof, I am posting it here wanting to know if it is valid. (It may not be as elegant as the textbook's proof though.)

(1) We do not consider the cases of $S \subseteq T$, $T \subseteq S$ or $S \cap T =...

Set Equivalence Problem]]>

Please see the attachment ,I couldn't write the question properly, and this is only one question but I need help with another one too.]]>

Functions problem]]>

- I want to describe in words the following sets:

1. $A:=\{(x,y)\in \mathbb{R}^2\mid x>0, y\leq 1\}$

$A$ is the set of all pointgs where the first coordinate is positiv and the second one is less or equal to $1$.

It is the subarea of the plane that is under the point $(0/1)$ to the right, without the y-axis.

Is this description enough or can we say also something else?

The graphical representation is:

2...

Description of sets]]>

- Show that the set of all positive rational numbers is a countable set.

(Hint: Consider all points in the first quadrant of the plane of which the coordinates x and y are integers.) - Show that the union of a countable number of countable sets is a countable set.

I have done the following:

- Let $x,y>0$. We write a positive rational $\frac{x}{y}$ at the point (x, y) in the plane in the first quadrant. Now we have to count these points, or...

Countable sets]]>

I am looking at an exercise that is formulated as follows:

Finite number k of squares on a circular route. The whole fuel in all is enough for 1 circle.

Show that there is a way to integrate the circle however the squares and the fuel are distributed.

There is also the following picture:

I haven't really understood the exercise statement. Could you explain that to me? What exactly do I have to show? ]]>

I am looking at the following problem:

There are $k$ couples and the host and the hostess. (So, in total $k+1$ couples.)

At the general greeting some people shake hands, others do not. Of course, nobody shakes hands with themselves or their spouse.

The host asks everyone else in the room how many people they shook hands with, and recieves $2k+1$ different answers.

How many hands did the hostess shake?

There are in total $2k+2$ people in the room (the guests and the host and...

Handshake Problem - Machines]]>

where B(p) = Printer p is busy and Q(j) = Print job j is queued.

When it's translated to symbol, we'll have (∀pB(p)) → (∃jQ(j)).

I'm trying to translate this statement to both English and symbol forms for

Following is what i have got so far:

in words: If there is a job in the queue, then every printer is busy.

in symbol: (∃jQ(j)) → (∀pB(p))...

Converse, Contrapositive and Negation for multiple Quantifiers]]>

I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.4 Ordinals ...

I need some help in fully understanding Theorem 1.4.6 ...

Theorem 1.4.6 reads as follows:

My question regarding the above proof by Micheal Searcoid is as follows:

How do we know that \(\displaystyle \alpha\) and...

Ordinals ... Searcoid, Theorem 1.4.6 ...]]>

I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.4 Ordinals ...

I need some help in fully understanding the Corollary to Theorem 1.4.4 ...

Theorem 1.4.4 and its corollary read as follows:

Searcoid gives no proof of Corollary 1.4.5 ...

To prove Corollary 1.4.5 we need to show \(\displaystyle \beta \in \alpha...\)

Ordinals ... Searcoid, Corollary 1.4.5 ... ...]]>

I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.4 Ordinals ...

I have another question regarding the proof of Theorem 1.4.4 ...

Theorem 1.4.4 reads as follows:

In the above proof by Searcoid we read the following:

"... ... Moreover, since \(\displaystyle x \subset \alpha\), we have \(\displaystyle \delta \in \alpha\). But...

Proper Subsets of Ordinals ... ... Another Question ... ... Searcoid, Theorem 1.4.4 ... ...]]>

I need to complete this question for an assignment, but I cannot seem to understand how to simplify the compound proposition with logical equivalences. If anyone here understands how to complete this question, please could you show me how, as it would be greatly appreciated. Thank you.

Here is the question: ]]>

I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.4 Ordinals ...

I need some help in fully understanding Theorem 1.4.4 ...

Theorem 1.4.4 reads as follows:

In the above proof by Searcoid we read the following:

"... ... Now, for each \(\displaystyle \gamma \in \beta\) , we have \(\displaystyle \gamma \in \alpha\) by 1.4.2, and the...

Proper Subsets of Ordinals ... ... Searcoid, Theorem 1.4.4 ... ...]]>

I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.3 Ordered Sets ...

I need some help in fully understanding some remarks by Searcoid following Definition 1.3.10 ...

Definition 1.3.10 and the remarks following read as follows:

In Searcoid's remarks following Definition 1.3.10 we read the...

Well Orders and Total Orders ... Searcoid Definition 1.3.10 ...]]>

I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.4 Ordinals ...

I need some help in fully understanding Theorem 1.4.3 ...

Theorem 1.4.3 reads as follows:

In the above proof by Searcoid we read the following:

"... ... Then \(\displaystyle \beta \subseteq \alpha\) so that \(\displaystyle \beta\) is...

Properties of the Ordinals ...]]>

I am currently focused on understanding Chapter 1: Sets ... and in particular Section 1.3 Ordered Sets ...

I need some help in fully understanding Theorem 1.3.24 ...

Theorem 1.3.24 reads as follows:

In the statement of Theorem 1.3.24 we read the following:

" ... ... Suppose that \(\displaystyle r\) is a function into \(\displaystyle X\) whose domain satisfies the hypothesis...

The Recursion Theorem ... Searcoid, Theroem 1.3.24 ... ...]]>

We have the sequence $$0, \ 2 , \ -6, \ 12, \ -20, \ \ldots$$ Its recursive definition is \begin{align*}&a_1=0 \\ &a_{n+1}=(-1)^{n+1}\cdot (a_n+2\cdot n)\end{align*} or not?

How can we convert that in the explicit form? ]]>

n

sigma 3i + 1 = n/2 (3n + 5)

i = n]]>

1. (∃x) ¬Fx Assumption

2. ¬Fy Assumption

3. (∀x) Fx Assumption

4. Fy 3, UI

5. (∀x) Fx ⇒ Fy 3-4, CP

6. ¬(∀x) Fx 5,2 MT

7. ¬(∀x) Fx...

Help with logic proof]]>

We have the letters 'WINPRESENT'. I want to calculate the rearrangements of these letters that contain either the word 'WIN' or the word 'PRESENT' or both of them.

I have done the following:

The subword 'WIN' is contained in $8!$ rearrangements. Since at 'PRESENT' we have twice the letter E we get $\frac{8!}{2}$ rearrangements. The subword 'PRESENT' is contained in $4!$ rearrangements. We have calculated twice 'WINPRESENT' and twice 'PRESENTWIN'.

So the total amount of...

Number of rearrangements of letters]]>

~ Ex Ay Ez P(x,y,z), where

~ is negation symbol, E existential quantifier, A universal quantifier.

We are asked to prove formally that, after instantiation of x to a, we obtain:

~ Ay Ez P(a,y,z)

How might we go about this? I appreciate all help,

agapito

Let $G$ be a graph of which the vertices are the permutations of $\{1,2,3,4,5,6,7,8,9,9,9\}$ with the property that two vertices $(\epsilon_1, \epsilon_2, \ldots, \epsilon_{11})$, $(\epsilon_1', \epsilon_2', \ldots, \epsilon_{11}')$ are connected with an edge if and only if the one is resulted from the other by exchanging the positions of two different integers.

How can we calculate the number of edges of the graph $G$ ?

We have $\frac{11!}{3!}$ (because we have 11 numbers...

Number of edges of the graph]]>

Wo consider the graph $G$ that we get by deleting any edge from the complete bipartite graph $K_{7,8}$. How many spanning trees does the complement graph $\overline{G}$ of $G$ have?

Could you give me a hint how we can find the desired number of spanning trees? ]]>