Question

Using properties of sets, show that: $A \cup (A \cap B) = A$

Answer 1

we have to show that $A\cup(A\cap B)=A$

proof :- we know, every set is a subset of itself.
e.g., $A\subset A$

we also know, $A\cap B$ is the set of all distinct elements that are in both A and B.
so, $A\cap B$ is a subset of A or B.
e.g., $A\cap B\subset A$
then definitely $A\cup(A\cap B)\subset A$......(1)
also we have, $A\subset A\cup(A\cap B)$.......(2)

from equations (1) and (2),
we know, it happens only when,
$A\cup(A\cap B)=A$

[ note : - property is if $A\subset B$, $B\subset A$ then, A = B]