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 $x\notin B$ 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