Using properties of sets prove the statements given in Exercise for all sets A and B, A ∪ (B

Home » Using properties of sets prove the statements given in Exercise for all sets A and B, A ∪ (B – A) = A ∪ B.

Using properties of sets prove the statements given in Exercise for all sets A and B, A ∪ (B – A) = A ∪ B.

Using properties of sets prove the statements given in Exercise for all sets A and B, A ∪ (B – A) = A ∪ B.

Solution:

As per the question,

Two sets A and B

We need to prove: $A \cup(B-A)=A \cup B$

Left Hand Side $=A \cup(B-A)$

As, $A-B=A \cap B^{\prime}$ , we have,

$=A \cup\left(B \cap A^{\prime}\right)$

As, distributive property of set $\Rightarrow(A \cup B) \cap(A \cup C)=A \cup(B \cap C)$ , we have,

$=(A \cup B) \cap\left(A \cup A^{\prime}\right)$

As, $A \cup A^{\prime}=U$ , we have,

$=(A \cup B) \cap U$ $=A \cup B$ $=$ Right Hand Side
As a result, hence proved

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