Answer to Question #126708 in Discrete Mathematics for kavee

Question #126708
11. Show that if A and B are sets, then
(a) A − B = A ∩ B
(b) (A ∩ B) ∪ (A ∩ B) = A
1
Expert's answer
2020-07-27T18:06:35-0400

a) AB=ABCA-B=A\cap B^C

We must show ABABCA-B\sube A\cap B^C and ABCABA\cap B^C\sube A-B

Show that ABABCA-B\sube A\cap B^C

Let xAB.x\in A-B. By definition of set difference, xAx\in A and xB.x\notin B. By definition of complement, xBx\notin B implies that xBC.x\in B^C. Hence, it is true that both, xAx\in A and xBC.x\in B^C. By definition of intersection, xABC.x\in A\cap B^C.

Show that ABCAB.A\cap B^C\sube A-B.

Let xABC.x\in A\cap B^C. By definition of intersection, xAx\in A and xBC.x\in B^C. By definition of complement, xBCx\in B^C implies that xB.x\notin B. Hence, xAx\in A and xB.x\notin B. By definition of set difference, xAB.x\in A-B.

Thus, AB=ABC.A-B=A\cap B^C.


b) (AB)(AB)=A(A-B)\cup(A\cap B)=A

Let xA.x\in A. There are two cases, xAx\in A and xBx\notin B or xAx\in A and xB.x\in B.

In the first case xA,xB,x\in A, x\notin B, so by definition set difference, xAB.x\in A-B.

In the second case xAx\in A and xB,x\in B, so by definition of intersection xAB.x\in A\cap B.

By definition of union x(AB)(AB).x\in (A-B)\cup(A\cap B).

Thus if xA,x\in A, then x(AB)(AB).x\in (A-B)\cup(A\cap B).


Let x(AB)(AB).x\in (A-B)\cup(A\cap B). This means that either xABx\in A-B or xAB.x\in A\cap B.

In the first case xA,xB,x\in A, x\notin B, in the second case xAx\in A and xB.x\in B.

Then in either case xA.x\in A.

Two sets are equal, since they have the same elements.Therefore


(AB)(AB)=A(A-B)\cup(A\cap B)=A

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