Question

If \( A = \{1, 2\} \) and \( B = \{3, 4, 5\} \), then find all number of relations from A to B.

Hint:

Remember the distinction: a relation is a subset of the Cartesian product, while a function is a special type of relation where each element of the first set is paired with exactly one element of the second set. The number of relations is always \( 2^{n(A)n(B)} \), while the number of functions is \( n(B)^{n(A)} \).

Answer 1

Step 1: Understanding the Concept:
A relation from a set A to a set B is any subset of the Cartesian product \( A \times B \). The total number of possible relations is therefore equal to the total number of possible subsets of \( A \times B \).
Step 2: Key Formula or Approach:
1. Let \( n(A) \) be the number of elements in set A and \( n(B) \) be the number of elements in set B.
2. The number of elements in the Cartesian product \( A \times B \) is \( n(A \times B) = n(A) \times n(B) \).
3. The number of subsets of a set with \( k \) elements is \( 2^k \).
4. Therefore, the number of relations from A to B is \( 2^{n(A) \times n(B)} \).
Step 3: Detailed Explanation or Calculation:
Given the sets:
\( A = \{1, 2\} \), so the number of elements in A is \( n(A) = 2 \).
\( B = \{3, 4, 5\} \), so the number of elements in B is \( n(B) = 3 \).
First, calculate the number of elements in the Cartesian product \( A \times B \):
\[ n(A \times B) = n(A) \times n(B) = 2 \times 3 = 6 \] A relation is a subset of \( A \times B \). The total number of subsets of a set with 6 elements is \( 2^6 \).
\[ \text{Number of relations} = 2^6 = 64 \] Step 4: Final Answer:
The total number of relations from A to B is 64.