Discrete Math Set Proofs Please solve number 5 Prove that (A union B)^c subsetof

Question

Discrete Math Set Proofs

Please solve number 5

Prove that (A union B)^c subsetofequalto A^c intersection B^c. Pf//Let ___ and __ be sets. Suppose that x 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 intersection (B - A) = phi Prove the given statement using the element method for proving that a set equals the empty set. Use the properties in Theorem 6.6.2to prove the given statement. Be sure to give a reason for every step.

Explanation / Answer

(3) Let A and B be sets.

Suppose that x (A U B)c [ We must show that x Ac Bc ]

Then by definition of compliment, x (A U B)

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

Consecuently, 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 compliment, x Ac and x Bc. Thus by definition of intersection, x Ac Bc. [as was to be shown].

(4) Let x be ANY element in A.

By definition, x B-A.

By definition, x A (B-A), if x A and x (B-A)

Since x B-A,

x A B-A

Since x is any element of A, this is true for all elements of A.

Thus A B-A = .

(For 5, please specify what the given statement is)

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