Q. 75.0( 2 Votes )

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.

Answer :

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.


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