Question

Use Use an element argument to prove the state (A-B)U(ANB) = A" is true. he statement "For all sets A and B,

Answer 1

Answer with Step-by-step explanation:

We have to prove that

`(A-B)\cup (A\cap B)=A` is true for all sets A and B

Let
`x\in(A-B)\cup (A\cap B)`

Then
`x\in(A-B)` or
`x\in(A\cap B)`

`x\in A` and
`x\\otin B` or
`x\in A\;and\; x\in B`

Hence,
`x\in A`

Conversely ,Let
`x\in A`

Then x belongs to A and x does not belongs to B then

x belongs to A- B

Or x belongs to A and x belongs to B then x belongs to
`A \cap B`

Hence,
`x\in( A-B)\cup (A\cap B)`

Therefore,
`(A-B)\cup (A\cap B)=A`

Answer 2

Answer with explanation:

To Prove: (A -B) ∪ (A ∩ B)=A

Proof:

Consider two sets A and B

⇒⇒ To prove A=B, in sets

We need to prove

A ⊆ B

and , B ⊆ A

then , A=B.

→A - B ⊆ A and A ∩ B ⊆ A

So, there are two possibilities

Either,→A⊆ A-B ∪ (A ∩ B) --------(1)

Or,→ A -B ∪ (A ∩ B)⊆ A-----------(2)

Combining (1) and (2) , we get

(A -B) ∪ (A ∩ B)=A