(a) Prove the following statement using the element method for proving that a set equals the empty set: For all sets A and B, A ? (B ? A) = ?. (b) Use the properties in Theorem 6.2.2 to prove the statement in part (a). Be sure to give a reason for every step.

Question

What will be an ideal response?

Answer 1

a. Proof: Suppose the given statement is false.
[We must show that this supposition leads logically to a contradiction.]
Then there exist sets A and B such that A ? (B ? A) ?= ?.
Thus there is an element, say x, in A ? (B ? A).
By definition of intersection, x ? A and x ? B ? A.
By definition of set difference (since x ? B ? A), x ? B and x /? A.
Hence x ? A and x /? A, which is a contradiction.
[Arriving at this contradiction shows that the given statement is not false; so it is true that for
all sets A and B, A ? (B ? A) = ?.]
b. Proof:
Let A and B be any sets. Then the left-hand side of the equation to be shown is
A ? (B ? A) = A ? (B ? Ac) by the set difference law
= (A ? B) ? Ac) by the associative law for ?
= (B ? A) ? Ac) by the commutative law for ?
= B ? (A ? Ac) by the associative law for ?
= B ? ? by the complement law for ?
= ? by the universal bound law for ? .
which is the right-hand side of the equation to be shown. [Hence the given statement is true.]