Discrete Math Disproofs, Algebraic Proofs, and Boolean Algebras Prove that (A B)

Question

Discrete Math

Disproofs, Algebraic Proofs, and Boolean Algebras

Prove that (A B)^C C A^c B^e. Pf//Let and besets. Suppose that .v elementof. [We must show x elementof _____] Then by definition of complement. So by definition of union, it is not the case that (x is in A or x is in B). Consequently, x is not in A __ x is not in B because of De Morgan's law of logic. In symbols, this says that and So by definition of complement, x elementof and x elementof ______. Thus, by definition of intersection, x elementof _____. [as was to be shown]. For all sets A and B, A (B - A) = theta (4) Prove the given statement using the element method for proving that a set equals the empty set. (5) Use the properties in Theorem 6.6.2to prove the given statement. Be sure to give a reason for every step.

Explanation / Answer

(3). Pf/ Let A and B be sets.

   Suppose that x AUB. [We must show that x Ac Bc ].

Then, by definition of complement, x A U B.

So, by definition of union, it is not the case that ( x is in A or x is in B)

   Consequently, x is not in A and x is not in B because of De Morgan’s law of logic.

In symbols, this says that x A and x B.

So, by definition of complement, x Ac and x Bc. Thus, by definition of intersection, x Ac Bc (as     was to be shown). This means that (AUB)c Ac Bc.

4. Pf/ Let A and B be sets.

Suppose that x A (B-A).

So, by the definition of intersection, x A and x (B-A).

Then, by the definition of (B-A), x B and x A.

However, this is a contradiction as x A. Therefore, A (B-A) = .

5. We do not have theorem 6.6.2

  

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