Question

these are my tries and other ppl's tries.

Prove these statements for any sets A & B. Prove using set definitions, or set equality property.

the question is in red, other information is my attempts.

Answer 1

• $A=B$ means the elements of both the sets are identical. i.e All elements of A are also elements of B and all elements in B are elements of A too. So if
  A B
  , then
  • Either at least one element of A is missing in B
  • Or at least one element of B is missing in A

This proves the statement that   

A B(E A, B) V ( A, xE B

Similarly, if an element present in one set is missing in the other set, the sets are not equal by definition. This proves the statement that  .

A B( E A, B) V ( A, xE B

Combining both the statement .

A B(E A, B) V ( A, xE B

• The statement to be proved is
  AUBCANB= A = B]
  

can be written as the union of three different parts as,

AUB

AUB (AB)U (B\ A) U (AnB
---------------------- (1)

where
A\B)
is that part of A which is absent in B and
(B A)
is that part of B which is absent in A.

Since it is given that
AUB
is a subset of $A \cap B$, Equation (1) says,

$(A\setminus B)\cup (B \setminus A)\cup(A \cap B) \subseteq A \cap B$

(AB)U (B A) =

.

(AB=(B\ A) A= B

Hence the statement is proved.