Answer with Step-by-step explanation:
We have to prove that
[tex] (A-B)\cup (A\cap B)=A[/tex] is true for all sets A and B
Let [tex] x\in(A-B)\cup (A\cap B)[/tex]
Then [tex] x\in(A-B)[/tex] or [tex] x\in(A\cap B)[/tex]
[tex] x\in A[/tex] and [tex] x\notin B[/tex] or [tex] x\in A\;and\; x\in B[/tex]
Hence, [tex] x\in A[/tex]
Conversely ,Let [tex] x\in A[/tex]
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 [tex] A \cap B[/tex]
Hence, [tex] x\in( A-B)\cup (A\cap B)[/tex]
Therefore, [tex] (A-B)\cup (A\cap B)=A[/tex]
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
