If A = {4n + 2 | n is a natural number} and B = .{3n | n is a natural number}, then what is (A ∩ B) equal to?a){12n2 + 6n | n is a natural number}b){24n -12 | n is a natural number}c){60n + 30 | n is a natural number}d){12n - 6 | n is a natural number}Correct answer is option 'D'. Can you explain this answer?

Question

If A = {4n + 2 | n is a natural number} and B = .{3n | n is a natural number}, then what is (A &cap; B) equal to?a){12n2 + 6n | n is a natural number}b){24n -12 | n is a natural number}c){60n + 30 | n is a natural number}d){12n - 6 | n is a natural number}Correct answer is option 'D'. Can you explain this answer?

Answer

Since, A = {An + 2 | n e N}
= {6,10,14,18,22,26,30,.,.}
and B = {3n | n ∈ N}
= {3,6,9,12,15,18,21,24,...}
So, A ∩ B = { 6 ,18,30,...}
= {6 + (n - 1)12 | n ∈ N}
= {12n - 6 | n ∈ N}
(c) x = 3(mod 7) => x - 3 = Ip, (p ∈ z) implies x= 7p + 3,p ∈ z i.e., {7p + 3 :p ∈ z}

Answer

To find the intersection of sets A and B, we need to find the common elements that appear in both sets.

Set A is defined as {4n + 2 | n is a natural number}. This means that for every natural number n, the element 4n + 2 is included in set A. For example, if n = 1, 4n + 2 = 4(1) + 2 = 6, so 6 is an element of set A.

Set B is defined as {3n | n is a natural number}. This means that for every natural number n, the element 3n is included in set B. For example, if n = 2, 3n = 3(2) = 6, so 6 is an element of set B.

To find the intersection of sets A and B, we need to find the elements that appear in both sets. In this case, we can see that the element 6 appears in both sets. Therefore, the intersection of sets A and B is {6}.

Answer

Since, A = {An + 2 | n e N}
= {6,10,14,18,22,26,30,.,.}
and B = {3n | n ∈ N}
= {3,6,9,12,15,18,21,24,...}
So, A ∩ B = { 6 ,18,30,...}
= {6 + (n - 1)12 | n ∈ N}
= {12n - 6 | n ∈ N}
(c) x = 3(mod 7) => x - 3 = Ip, (p ∈ z) implies x= 7p + 3,p ∈ z i.e., {7p + 3 :p ∈ z}

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