We are given two sets $A = \{a, b, c, d\}$ and $B = \{1, 2, 3, 4, 5\}$. We need to determine: i) The number of functions from $A$ to $B$. ii) The number of functions from $B$ to $A$.

1. Problem Description

We are given two sets $A = \{a, b, c, d\}$ and $B = \{1, 2, 3, 4, 5\}$. We need to determine:

i) The number of functions from $A$ to $B$.

ii) The number of functions from $B$ to $A$.

2. Solution Steps

i) The number of functions from $A$ to $B$.

The cardinality of set $A$ is $|A| = 4$, and the cardinality of set $B$ is $|B| = 5$.

The number of functions from a set $A$ to a set $B$ is given by $|B|^{|A|}$.

In this case, the number of functions from $A$ to $B$ is $5^4$.

$5^4 = 5 \times 5 \times 5 \times 5 = 25 \times 25 = 625$.

ii) The number of functions from $B$ to $A$.

The cardinality of set $A$ is $|A| = 4$, and the cardinality of set $B$ is $|B| = 5$.

The number of functions from a set $B$ to a set $A$ is given by $|A|^{|B|}$.

In this case, the number of functions from $B$ to $A$ is $4^5$.

$4^5 = 4 \times 4 \times 4 \times 4 \times 4 = 16 \times 16 \times 4 = 256 \times 4 = 1024$.

3. Final Answer

i) The number of functions from $A$ to $B$ is

6

2

5. ii) The number of functions from $B$ to $A$ is 1024.