Question 8 of 33intermediate🔧 ApplyLong Answer4 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

Step 1: Finding A×BA \times B

Given A={1,2}A = \{1, 2\} and B={3,4}B = \{3, 4\}.

A×BA \times B is the set of all ordered pairs (x,y)(x, y) such that x∈Ax \in A and y∈By \in B.

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

Step 2: Number of Subsets

A×BA \times B has 4 elements.

Number of elements in A×B=n(A)×n(B)=2×2=4A \times B = n(A) \times n(B) = 2 \times 2 = 4

The number of subsets of a set with nn elements =2n= 2^n

Therefore, number of subsets of A×B=24=A \times B = 2^4 = 16 subsets

Step 3: List of All Subsets

The 16 subsets of A×BA \times B are:

  1. ϕ\phi (empty set)
  2. {(1,3)}\{(1, 3)\}
  3. {(1,4)}\{(1, 4)\}
  4. {(2,3)}\{(2, 3)\}
  5. {(2,4)}\{(2, 4)\}
  6. {(1,3),(1,4)}\{(1, 3), (1, 4)\}
  7. {(1,3),(2,3)}\{(1, 3), (2, 3)\}
  8. {(1,3),(2,4)}\{(1, 3), (2, 4)\}
  9. {(1,4),(2,3)}\{(1, 4), (2, 3)\}
  10. {(1,4),(2,4)}\{(1, 4), (2, 4)\}
  11. {(2,3),(2,4)}\{(2, 3), (2, 4)\}
  12. {(1,3),(1,4),(2,3)}\{(1, 3), (1, 4), (2, 3)\}
  13. {(1,3),(1,4),(2,4)}\{(1, 3), (1, 4), (2, 4)\}
  14. {(1,3),(2,3),(2,4)}\{(1, 3), (2, 3), (2, 4)\}
  15. {(1,4),(2,3),(2,4)}\{(1, 4), (2, 3), (2, 4)\}
  16. {(1,3),(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, A×BA \times B is defined as 'a non-empty set of ordered pairs (x,y)(x, y) such that x∈Ax \in A and y∈By \in B'. The example in the context shows how ordered pairs are formed systematically.

For subsets, the power set formula 2n2^n applies where nn is the number of elements. The question is from Exercise 2.1 and tests both computation and enumeration skills.

Solution Steps

  1. Step 1: Form A×BA \times B by pairing each element of AA with each element of BB

  2. Step 2: Count elements in A×B=4A \times B = 4, so number of subsets =24=16= 2^4 = 16

  3. Step 3: List all subsets starting from empty set, singletons, pairs, triplets, to the full set

PreviousNext