Question 6 of 33intermediate🔧 ApplyShort Answer2 marks

If A×B={(a,x),(a,y),(b,x),(b,y)}A \times B = \{(a, x), (a, y), (b, x), (b, y)\}. Find AA and BB.

Correct Answer

A = {a, b} and B = {x, y}.

The Cartesian product A × B is defined as the set of all ordered pairs (x, y) such that x ∈ A and y ∈ B. In the given set A × B = {(a, x), (a, y), (b, x), (b, y)}, the first elements of all ordered pairs are 'a' and 'b'. Therefore, A = {a, b}.

Similarly, the second elements of all ordered pairs are 'x' and 'y'. Therefore, B = {x, y}. This matches the pattern where A is the set of first elements and B is the set of second elements from the ordered pairs in the Cartesian product.

Exercise: EXERCISE 2.1 | Q: 6 | (Page 4)
For More Understanding

Explanation

This question is directly solved using the method shown in Example 6 of the textbook. The key concept is that in a Cartesian product A × B, set A consists of all distinct first elements of the ordered pairs, and set B consists of all distinct second elements. The textbook example demonstrates this with A × B = {(p, q), (p, r), (m, q), (m, r)} yielding A = {p, m} and B = {q, r}.

Solution Steps

  1. Step 1: Identify all first elements from the ordered pairs in A × B = {(a, x), (a, y), (b, x), (b, y)}. The first elements are: a, a, b, b.

  2. Step 2: Form set A from the distinct first elements. Therefore, A = {a, b}.

  3. Step 3: Identify all second elements from the ordered pairs. The second elements are: x, y, x, y.

  4. Step 4: Form set B from the distinct second elements. Therefore, B = {x, y}.

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