Question

Let $A = \{1, 2, 3\}$ and $B = \{a, b\}$ then number of relations from set $A$ to set $B$ are

• A. 64
• B. 8
• C. 4
• D. 6

Hint:

Distinct counting formulas to remember:
- Number of relations from $A$ to $B = 2^{m n}$
- Number of functions from $A$ to $B = n^m$
- Number of one-to-one (injective) functions $= P(n, m) = \frac{n!}{(n-m)!}$ (for $n \geq m$)

Answer 1

The Correct Option is A

Step 1: Concept
In set theory, a relation $R$ from a set $A$ to a set $B$ is defined as any subset of the Cartesian product $A \times B$, i.e., $R \subseteq A \times B$.
The Cartesian product $A \times B$ consists of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$.

Step 2: Key Formulas and Approach
Let $|A| = m$ and $|B| = n$ denote the cardinalities of sets $A$ and $B$, respectively.
1. The total number of ordered pairs in $A \times B$ is given by: \[ |A \times B| = |A| \cdot |B| = m \cdot n \] 2. Since any subset of $A \times B$ represents a valid relation, the total number of relations from $A$ to $B$ equals the number of elements in the power set $\mathcal{P}(A \times B)$: \[ \text{Total number of relations} = 2^{|A \times B|} = 2^{m \cdot n} \]

Step 3: Step-by-step Explanation

• Count the number of elements in set $A = \{1, 2, 3\}$: \[ m = |A| = 3 \]
• Count the number of elements in set $B = \{a, b\}$: \[ n = |B| = 2 \]
• Calculate the total number of elements in the Cartesian product $A \times B$: \[ |A \times B| = 3 \times 2 = 6 \]
• The elements of $A \times B$ explicitly are: \[ A \times B = \{(1, a), (1, b), (2, a), (2, b), (3, a), (3, b)\} \]
• Compute the total number of subsets of $A \times B$: \[ \text{Total relations} = 2^6 = 64 \]

Step 4: Final Answer
The total number of relations from set $A$ to set $B$ is $64$. Thus, Option (A) is correct.