Question

If \( A = \{a, b, c\} \) and \( B = \{1, 2\} \), then find the number of relations from \( A \) to \( B \).

Hint:

The number of relations between two sets \( A \) and \( B \) is equal to \( 2^{|A \times B|} \), where \( |A \times B| \) is the number of elements in the Cartesian product \( A \times B \).

Answer 1

A relation from set \( A \) to set \( B \) is a subset of the Cartesian product \( A \times B \). The Cartesian product \( A \times B \) consists of all possible ordered pairs \( (a, b) \), where \( a \in A \) and \( b \in B \). For the given sets: \[ A = \{a, b, c\}, B = \{1, 2\}, \] the Cartesian product \( A \times B \) contains: \[ A \times B = \{(a, 1), (a, 2), (b, 1), (b, 2), (c, 1), (c, 2)\}. \] This gives us 6 elements in \( A \times B \).

Step 1: Number of relations.
A relation is any subset of \( A \times B \). The number of subsets of a set with \( n \) elements is \( 2^n \). Here, \( A \times B \) has 6 elements, so the number of relations is: \[ 2^6 = 64. \] Conclusion: The number of relations from \( A \) to \( B \) is \( \boxed{64} \).