Question 8 of 35intermediate🔧 ApplyShort Answer3 marks

Let A={1,2}A = \{1, 2\} and B={3,4}B = \{3, 4\}. Write A×BA \times B. How many subsets will A×BA \times B have? List them.

Correct Answer

A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}

Since A = {1, 2} has 2 elements and B = {3, 4} has 2 elements, the Cartesian product A × B contains 2×22 \times 2 = 4 ordered pairs.

Number of subsets = 2⁴ = 16

The 16 subsets of A × B are: φ, {(1,3)}, {(1,4)}, {(2,3)}, {(2,4)}, {(1,3),(1,4)}, {(1,3),(2,3)}, {(1,3),(2,4)}, {(1,4),(2,3)}, {(1,4),(2,4)}, {(2,3),(2,4)}, {(1,3),(1,4),(2,3)}, {(1,3),(1,4),(2,4)}, {(1,3),(2,3),(2,4)}, {(1,4),(2,3),(2,4)}, {(1,3),(1,4),(2,3),(2,4)}

Exercise: EXERCISE 2.1 | Q: 8 | (Page 4)
For More Understanding

Explanation

This question tests understanding of Cartesian products and subsets. From the context, we see that A × B consists of all ordered pairs (a, b) where a ∈ A and b ∈ B. For A = {1, 2} and B = {3, 4}, we pair each element of A with each element of B. The number of elements in A × B equals n(A) × n(B) = 2×22 \times 2 = 4. A set with n elements has 2n2^{n} subsets, so A × B has 2⁴ = 16 subsets including the empty set and the set itself.

Solution Steps

  1. Step 1: Find A × B by pairing each element of A with each element of B: {(1,3), (1,4), (2,3), (2,4)}

  2. Step 2: Count elements: n(A × B) = n(A) × n(B) = 2×22 \times 2 = 4

  3. Step 3: Calculate number of subsets: 2⁴ = 16

  4. Step 4: List all 16 subsets systematically - starting from empty set, singletons, pairs, triplets, and finally the complete set

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