Find Total Number of Functions from \(A\) to \(B\)

📝 Question

Let:

\[ A=\{1,2,3\}, \quad B=\{a,b\} \]

Find the total number of functions from \(A\) to \(B\).

✅ Solution

🔹 Step 1: Use Formula

If a set \(A\) has \(m\) elements and set \(B\) has \(n\) elements, then:

\[ \text{Number of functions} = n^m \]

(Each element of \(A\) has \(n\) choices in \(B\))

🔹 Step 2: Substitute Values

\[ |A|=3,\quad |B|=2 \]

\[ \text{Number of functions} = 2^3 \] —

🔹 Step 3: Calculate

\[ 2^3 = 8 \] —

🎯 Final Answer

\[ \boxed{8} \]

🚀 Exam Shortcut

• Formula: \(n^m\)
• Domain size = power
• Codomain size = base
• Here: \(2^3 = 8\)