Question

Statement I: Let C and D be two finite sets having 3 and 4 elements respectively. Then the number of functions from C to D is 81. Statement II: Every function from set C to set D is a subset of C×D.

• A. Both Statement I and Statement II are correct.
• B. Statement I is correct but Statement II is incorrect.
• C. Statement II is correct but Statement I is incorrect.
• D. Both Statement I and Statement II are incorrect

Answer 1

Answer: (C)

c. Statement II is correct but Statement I is incorrect.

Answer 2

Statement I Analysis:

Let $C$ and $D$ be two finite sets with $|C| = m$ and $|D| = n$. The number of functions from $C$ to $D$ is given by $n^m$.

In this case, $|C| = 3$ and $|D| = 4$. So, $m=3$ and $n=4$.

The number of functions from $C$ to $D$ is $4^3 = 4 \times 4 \times 4 = 64$.

The statement says the number of functions is 81. Since $64 \neq 81$, Statement I is incorrect.

Statement II Analysis:

A function $f: C \to D$ is a relation from $C$ to $D$ such that for every element $x \in C$, there exists a unique element $y \in D$ such that $(x, y)$ belongs to the relation.

A relation from $C$ to $D$ is defined as a subset of the Cartesian product $C \times D$. The Cartesian product $C \times D$ is the set of all ordered pairs $(c, d)$ where $c \in C$ and $d \in D$.

The function $f$ can be represented by its graph, which is the set of ordered pairs $\{(c, f(c)) \mid c \in C\}$.

For any $c \in C$, $f(c)$ is the unique element in D assigned to $c$. Thus, $f(c) \in D$.

Therefore, each ordered pair $(c, f(c))$ is an element of $C \times D$.

The set $\{(c, f(c)) \mid c \in C\}$, which represents the function $f$, is a subset of $C \times D$.

Thus, every function from set $C$ to set $D$ is a subset of $C \times D$. Statement II is correct.

Conclusion:

Statement I is incorrect. Statement II is correct.

This matches option (c).