Question

prove that (a u b) = x(a) + x(b) - x( a b ).​

Answer 1

Question:

Prove that $n(A\cup B)=n(A)+n(B)-n(A\cap B)$

Solution:

• $n(S)$ indicates the number of elements in the set S.
• $A\cup B$ is the set whose elements are also in the set A or also in the set B. It is the set containing all elements of the sets A and B.

$A\cup B=\{x:x\in A\lor x\in B\}$

• $A\cap B$ is the set whose elements are both in the set A and in the set B. It is the set containing common elements of the sets A and B.

$A\cap B=\{x:x\in A\land x\in B\}$

Let's come to the proof.

It is true that $n(A\cup B)=n(A)+n(B)$ if and only if both A and B are disjoint, i.e., $A\cap B=\phi.$ But in our case it should not be.

Well, we can have,

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

Thus,

$n(A\cup B)=n(A\cup(B-(A\cap B)))$

Since $A$ and $B-(A\cap B)$ are disjoint,

$n(A\cup B)=n(A)+n(B-(A\cap B))$

But $n(B-(A\cap B))=n(B)-n(A\cap B).$ Then,

$\boxed{n(A\cup B)=n(A)+n(B)-n(A\cap B)}$

Hence the Proof!

#answerwithquality

#BAL