For A = {x | x is a prime factor of 42}, B = {x | 5 < x 12, } and C = {1, 4, 5, 6}, verify A ∪ (B ∪ C) = (A ∪ B) ∪ C.

From the above statements we can find the elements of the set.

So,

A = {2, 3, 7}

B = {6, 7, 8, 9, 10, 11, 12}

C = {1, 4, 5, 6}

*Prime numbers are the numbers which can only be divided by 1 or the number itself.

*Factors are the number which are multiplied together to get a new number so the term prime factorization may be defined as the finding the prime numbers which are multiplied together to give the source word.

The elements of B are natural number and they lies between 6 and 12.

Here we have to verify A ⋃ (B ⋃ C) = (A ⋃ B) ⋃ C. It’s the associative union property of set theory.

L.H.S.

A ⋃ (B ⋃ C)

Here first we will split the entire operation and have to find elements of (B ⋃ C) using the given data then we will find the union of set A and the set obtained from the union of B and C

So (B ⋃ C) = {6, 7, 8, 9, 10, 11, 12} ⋃ {1, 4, 5, 6}

= {6, 7, 8, 9, 10, 11, 12, 1, 4, 5}

Now

A ⋃ (B ⋃ C) = {2, 3, 7} ⋃ {6, 7, 8, 9, 10, 11, 12, 1, 4, 5}

= {2, 3, 7, 6, 8, 9, 10, 11, 12, 1, 4, 5}

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}…………..(1)

R.H.S

(A ⋃ B) ⋃ C

Here first we have to find the value of (A ⋃ B) from the given data then we will find the union between the set obtained from (B ⋃ C)

And C

So (A ⋃ B) = {2, 3, 7} ⋃ {6, 7, 8, 9, 10, 11, 12}

= {2, 3, 7, 6, 8, 9, 10, 11, 12}

Now

(A ⋃ B) ⋃ C = {2, 3, 7, 6, 8, 9, 10, 11, 12} ⋃ {1, 4, 5, 6}

= {2, 3, 7, 6, 8, 9, 10, 11, 12, 1, 4, 5}

= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}…………..(2)

So from (1) & (2) it’s clear that L.H.S. = R.H.S which verifies that

A ⋃ (B ⋃ C) = (A ⋃ B) ⋃ C.