Proof Involving Binary Operators

In summary, the conversation is about proving the equality of two sets, A union (B intersection C) and (A union B) intersection (A union C), which is one of DeMorgan's Laws. The discussion involves considering two cases and using the associative property to rewrite one of the sets. The conversation also mentions using Venn diagrams to solve the problem, but it is ultimately solved without them by showing that the elements in one set are also in the other set.
  • #1
Caldus
106
0
I am trying to prove that A union (B intersection C) = (A union B) intersection (A union C). In other words, proving one of DeMorgan's Laws. I have gotten this far, and not sure if I'm right thus far:

Let x belong to A union (B intersection C). Then x is in either A or in (B intersection C).

Case 1: x belongs to A.
In this case, x belongs to (A union B) and x belongs to (A union C).

Case 2: x belongs to (B intersection C).
In this case, x belongs to either (A union B) or (A union C).

So x belongs to ((A union B) intersection (A union C)) union ((A union B) union (A union C)).

((A union B) union (A union C)) can be rewritten as (using associative property):

(A union B union A union C), or simply (A union B union C).

What do I do now? Thank you.
 
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  • #2
Case 2: x belongs to (B intersection C).
In this case, x belongs to either (A union B) or (A union C).

No, if x belongs to (B intersection C),then x belongs to both (A union B) and (A union C).
(since x belongs to (B intersection C) it is in both B and C and so in both (A union B) and (A union C) so it is in
(A union B)intersection (A union C).)
 
  • #3
OK this is what I got now:

First, we must prove that A union (B intersection C) is a subset of (A union B) intersection (A union C).

So we let x belong to A union (B intersection C). Then x is in A or (B intersection C).

Case 1: x belongs to A.
In this case, x belongs to (A union B) and x belongs to (A union C).

Case 2: x belongs to (B intersection C).
In this case, x belongs to (A union B) and x belongs to (A union C).

So x belongs to ((A union B) intersection (A union C)).

Now we have to prove that (A union B) intersection (A union C) is a subset of A union (B intersection C).

So now let x belong to (A union B) intersection (A union C). Then x is in A. x is in either B or C.

Case 1: x is in B.
In this case, x belongs to (A union B).

Case 2: x is in C.
In this case, x belongs to (A union C).

Am I right so far? I don't know how to go any farther with this.

Thanks for the help so far.
 
  • #4
Why not do it with Venn diagrams?
 
  • #5
I have to solve it without using Venn Diagrams.
 
  • #6
How can I conclude that (A union B) intersection (A union C) is a subset of A union (B intersection C)?
 
  • #7
almost any question about the containment or equality of sets boils down to showing x in A implies x in D

so take something in one and show it's in the other by considering all cases if necessary. here x is in AUB and AUC, if it is in A it is certainly in AU(B int C)

if it is not in A then it must be in both B and C, ie it is in (B int C), and is then also clearly in AU(BintC)
 

1. What are binary operators?

Binary operators are mathematical symbols or operations that take two operands, or values, and produce a new value. Examples of binary operators include addition (+), subtraction (-), multiplication (*), and division (/).

2. How are binary operators used in proofs?

Binary operators can be used in proofs to show the relationship between two values or to manipulate equations. For example, in a proof involving addition, you may use the binary operator + to show that two values are equal or to transform an equation into a different form.

3. What are some common properties of binary operators used in proofs?

Some common properties of binary operators used in proofs include commutativity, associativity, and distributivity. Commutativity means that the order of the operands does not affect the result (e.g. a + b = b + a). Associativity means that the grouping of operands does not affect the result (e.g. (a + b) + c = a + (b + c)). Distributivity means that the operator can be distributed over another operator (e.g. a * (b + c) = a * b + a * c).

4. Are there any rules for using binary operators in proofs?

Yes, there are some rules for using binary operators in proofs. One important rule is the order of operations, which states that some operators should be evaluated before others. For example, in the equation 2 + 3 * 4, the multiplication should be done before the addition. Another rule is that you must maintain equality on both sides of the equation when using binary operators.

5. Can binary operators be used in proofs outside of mathematics?

Yes, binary operators can be used in proofs outside of mathematics, such as in computer science and logic. In computer programming, binary operators are used to manipulate data and make decisions. In logic, binary operators are used to represent logical relationships, such as AND and OR.

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