Question:

The Cartesian product A x A has 9 elements among which are found (-1, 0) and (0, 1). Find the set A and the remaining elements of A x A.

Updated On: Oct 21, 2023
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Solution and Explanation

We know that if n(A) = p and n(B) = q, then n(A x B) = pq.
∴ n(A x A) = n(A) x n(A)
It is given that n(A x A) = 9
∴ n(A) x n(A) = 9
⇒ n(A) = 3
The ordered pairs (-1, 0) and (0, 1) are two of the nine elements of A x A.
We know that A x A = {(a, a): a ∈A}. Therefore, -1, 0, and 1 are elements of A.
Since n(A) = 3, it is clear that A = {-1, 0, 1}.
The remaining elements of set A x A are (-1, -1), (-1, 1), (0, -1), (0, 0), (1, -1), (1, 0), and (1, 1)

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Concepts Used:

Relations and functions

A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.

A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.

Representation of Relation and Function

Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.

  1. Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
  2. Roster form - {(1, 1), (2, 4), (3, 9)}
  3. Arrow Representation