Verify the following identities where A = { 1,2,3,4,5} , B = { 2,3,5,6} ,C = { 4,5,6,7} }}

${A \cap (B \cup C) = (A \cap B) \cup (A \cap C) }$

Hint- First let’s learn the meaning of symbols.

$\cap \to$ Intersection

$\cup \to$ Union

$X \cap Y$ Means only common elements of X and Y

$X \cup Y$ Means all elements of X and Y.

Complete answer:

Now let's verify by equating LHS = RHS

LHS,

First we find $(B \cup C)$ means we take all elements of B and C.

$\Rightarrow$ { 2,3,4,5,6,7}

Now we find $A \cap {\text{(}}B \cup C)$ means we take common $(B \cup C)$ and A.

$\Rightarrow$ { 2,3,4,5}

RHS,

First we find $(A \cap C)$ means to take common elements between A and C.

$\Rightarrow$ { 4,5}

Then, we find $(A \cap B)$ means to take common elements between A and B.

$\Rightarrow$ { 2,3,5}

Now, finally we find $(A \cap B) \cup {\text{(}}A \cap C)$ means we take all elements of $(A \cap C)$ and $(A \cap B)$

$\Rightarrow$ { 2,3,4,5}

Here , LHS = RHS proved.

Note: - Be careful with notations. We should select elements properly. If we make a single mistake in selection, we’ll get wrong answers in the end.