If two sets A and B are having 99 elements in common, then the number of elements common to each ofthe set A x B and B x A area)299b)992c)100d)18Correct answer is option 'B'. Can you explain this answer?

Question

Answer

Ans: 99²
because no. of elements common in A×B and B×A= square of no. of elements common in A and B.
for example: A= {1,2,3} and B= {2,3,4}
n(A intersection B)= 2
A×B= {(1,2),(1,3),(1,4),(2,2),(2,3),(2,4),(3,2),(3,3),(3,4)}
B×A= {(2,1),(2,2),(2,3),(3,1),(3,2),(3,3),(4,1),(4,2),(4,3)}
common= (2,3),(3,2),(3,3),(2,2)
thus no. of common = 4 = 2²

Answer

99²

Answer

Explanation:

To find the number of elements common to A x B and B x A, we need to understand the concepts of Cartesian product and set intersection.

Cartesian Product:
The Cartesian product of two sets A and B, denoted as A x B, is the set of all ordered pairs (a, b) where a belongs to A and b belongs to B.

For example, if A = {1, 2} and B = {a, b}, then A x B = {(1, a), (1, b), (2, a), (2, b)}.

Set Intersection:
The intersection of two sets A and B, denoted as A ∩ B, is the set of all elements that are common to both A and B.

For example, if A = {1, 2, 3} and B = {2, 3, 4}, then A ∩ B = {2, 3}.

Number of Elements Common to A x B and B x A:
Given that A and B have 99 elements in common, we can represent this as A ∩ B = {x1, x2, ..., x99}.

To find the number of elements common to A x B and B x A, we need to find the intersection of A x B and B x A.

A x B = {(a1, b1), (a1, b2), ..., (a1, bn), (a2, b1), (a2, b2), ..., (a2, bn), ..., (am, b1), (am, b2), ..., (am, bn)}
B x A = {(b1, a1), (b1, a2), ..., (b1, am), (b2, a1), (b2, a2), ..., (b2, am), ..., (bn, a1), (bn, a2), ..., (bn, am)}

To find the intersection of A x B and B x A, we need to find all ordered pairs that are common to both sets.

Since A ∩ B = {x1, x2, ..., x99}, the ordered pairs that are common to A x B and B x A will have the form (xi, xi) where xi belongs to A ∩ B.

Therefore, the number of elements common to A x B and B x A is 99.

Hence, the correct answer is option B) 992.

Answer

A cyclic group is always an abelian group but every abelian group is not a cyclic group. For instance, the rational numbers under addition is an abelian group but is not a cyclic one.

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