Given P = {a, b, c, d, e}, Q = {a, e, i, o, u} and R = {a, c, e, g}. Verify the associative property of set intersection.

The associative property of set intersection says that for three given sets the intersection operation between them, are associative in nature. Here for the three sets P, Q, R the associative property of set intersection can be represented as

P ⋂ (Q ⋂ R) = (P ⋂ Q) ⋂ R

So using the data given we have to prove the above statement.

Data given,

P = {a, b, c, d, e}

Q = {a, e, i, o, u}

R = {a, c, e, g}

L.H.S

For ease of solving we will split the statement P ⋂ (Q ⋂ R) in two part where first we use the intersection operation between Q and R then another intersection will be done between P and values obtained from intersection operation between Q and R.

So, (Q ⋂ R) = {a, e, i, o, u} ⋂ {a, c, e, g}

= {a, e}

Now, P ⋂ (Q ⋂ R) = {a, b, c, d, e} ⋂ {a, e}

= {a, e}…….(1)

R.H.S.

For ease of solving we will split the statement (P ⋂ Q) ⋂ R in two part where first we use the intersection operation between P and R then another intersection will be done between values obtained from intersection operation between P and Q and values of R.

So (P ⋂ Q) = {a, b, c, d, e} ⋂ {a, e, i, o, u}

= {a, e}

Now (P ⋂ Q) ⋂ R = {a, e} ⋂ {a, c, e, g}

= {a, e}……………… (2)

From statement (1) and (2) it’s clear that L.H.S is equal to R.H.S which proved that P ⋂ (Q ⋂ R) = (P ⋂ Q) ⋂ R.