# Question in Function

#### noof

##### New member
Hello

how are towdayyyyyy??

I have quistion

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Here file

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#### Plato

##### Well-known member
MHB Math Helper
Re: Qustion in Funiction

Hello

how are towdayyyyyy??

I have quistion

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In each of the following two sets, A and B, are given. Answer the following questions about each
pair of them.
(a) Is A = B?
(b) Is A  B?
(c) Is B  A?
(d) Compare the cardinalities of A and B.
(i) A = P(P(P()) B = P(P(P(P()))
(ii) A = P(X [ Y ) B = P(X) [ PY )
(iii) A = P(X \ Y ) B = P(X) \ P(Y )
(iv) A = P(X  Y ) B = P(X)  P(Y )

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Let X be a set and let f1 and f2 be functions from A to R. For x 2 X let g(x) = f1(x) + f2(x) and
h(x) = f1(x)f2(x). Verify that g and h are functions.

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In class we showed that jZ+j = jNj and that jZ􀀀j = jNj. Give a bijective function f : N ! Z to
show that jNj = jZj (i.e. cardinality of the set of natural numbers ([f0; 1; 2; 3; : : :g) is the same as the
cardinality of set of all integers).
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Using de nitions of the set operations show that if

X  Y and X  Z then X  Y \ Z

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Express the following in terms of predicate logic (using nested quanti er and appropriately de ned
predicates).
(a) If jXj < jY j, then there can not be an onto function from X to Y .
(b) If jXj > jY j, then there can not be an one-to-one function from X to Y .
(c) Principle of mathematical induction is an important proof technique which works as follows:
Suppose we want to show that the predicate P is true for all positive integers n, we complete
two steps.
 Basis step: Show that P is true for 1.
 Inductive step: Show that for every positive integer k, if P is true for k then P is true
for k + 1.
(d) The principle of Well Ordering states that \every nonempty set of positive integers has a
minimum element".
(e) The Pigeon-hole Principle states that if n+1 pigeons are placed in n pigeon-holes then some
pigeon-hole must contain more than 1 pigeons.
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pleeeeeeez help me any Q ........
If you expect any help, edit your post. Get rid of any special fonts and/or junk symbols.

#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
Re: Qustion in Funiction

Hello, and welcome to the forum!

I'd like to remind a couple of forum rules:
8. Do not ask too many questions in one thread. Do not ask more than two questions in a post.
11. Show some effort. If you want help with a question it is expected that you will show some effort. Effort might include showing your working, taking the time to learn how to typeset equations using LaTeX, formatting your question so that it is more easily understood, using effective post titles and posting in the appropriate subforum, making a genuine attempt to understand the help that is given before asking for more help and learning from previous questions asked.
It's also pretty difficult to understand your notation. Try writing your questions using LaTeX. Put formulas between two $\$$'s. For example, \$$ \emptyset \subseteq A^B$\ gives $\emptyset\subseteq A^B$ (JavaScript needs to be enabled). You can right-click on a formula to see its LaTeX source code. See this LaTeX tutorial on Wikibooks, especially here and here. You could also copy-paste mathematical Unicode symbols from this Wikipedia page, though using LaTeX is preferred. Finally, many mathematical symbols can be typed using plain text and English, e.g., f : A -> B, x is in A, A is a subset of B, etc.

Concerning a bijective function $f:\mathbb{N}\to\mathbb{Z}$, consider
$f(n)= \begin{cases} k&\text{when }n=2k\\ -k-1&\text{when }n=2k+1 \end{cases}$

#### noof

##### New member
Re: Qustion in Funiction

Here file

pleeeeeeeeeeeeez help me un any quistion

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#### Evgeny.Makarov

##### Well-known member
MHB Math Scholar
Re: Qustion in Funiction

It seems like this is homework. MHB policy (rule #6) is not to help with for-credit assignments. Sorry. However, you are still welcome to discuss concepts that you covered, e.g., bijection, induction, well-ordering principle, etc.

#### noof

##### New member
Thanks for all

I hope that anyone trying to understand this topic

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