Question

Using properties of sets, show that:

$A \cap(A \cup B) = B$
​

Answer 1

Answer:

LHS = A ∩ (A ∪ B)

Use distribution property,

e.g., P\cap(Q\cup R)=(P\cap Q)\cup(P\cap R)P∩(Q∪R)=(P∩Q)∪(P∩R)

= (A ∩ A) ∪ (A ∩ B)

Use relation A ∩ A = A

= (A ∩ A) ∪ (A ∩ B)

= A ∪ (A ∩ B)

= A = RHS

Hence, A ∩ (A ∪ B) = A

Step-by-step explanation:

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Answer 2

A ∩ (A ∪ B) = A

L.H.S

⇒ A ∩ (A ∪ B)

Using distribution property we get

⇒ (A ∩ A) ∪ (A ∩ B)

Use relation A ∩ A = A

⇒ (A ∩ A) ∪ (A ∩ B)

⇒ A ∪ (A ∩ B)

⇒ A

R.H.S

Hence, A ∩ (A ∪ B) = A