f and h are function from A `rarr` B, where A = {a, b, c, d} and B = {s, t, u} defined as follows
f(a) = t, f (b) = s, f(c ) = s
f(d) = u, h(a) = s, h(b) = t
h(c ) = s, h(a) = u, h(d) = u
Which one of the following statement is true?

Options

A. A. f and h are functions
B. B. f is a function and h is not a function
C. C. f and h are not functions
D. D. None of the above

Explanation

To determine which statement is true regarding the functions \( f \) and \( h \) defined from set \( A \) to set \( B \), we will analyze the provided definitions of both functions step by step.

### Step 1: Define the Sets and Functions
We have:
- Set \( A = \{a, b, c, d\} \)
- Set \( B = \{s, t, u\} \)

The functions are defined as follows:
- For function \( f \):
- \( f(a) = t \)
- \( f(b) = s \)
- \( f(c) = s \)
- \( f(d) = u \)

- For function \( h \):
- \( h(a) = s \)
- \( h(b) = t \)
- \( h(c) = s \)
- \( h(d) = u \)

### Step 2: Check if \( f \) is a Function
A relation is a function if every element in the domain (set \( A \)) maps to exactly one element in the codomain (set \( B \)).

- **Mapping for \( f \)**:
- \( a \) maps to \( t \) (1 image)
- \( b \) maps to \( s \) (1 image)
- \( c \) maps to \( s \) (1 image)
- \( d \) maps to \( u \) (1 image)

Since each element in \( A \) has exactly one image in \( B \), \( f \) is a function.

### Step 3: Check if \( h \) is a Function
Now, we will check if \( h \) is a function.

- **Mapping for \( h \)**:
- \( a \) maps to \( s \) (1 image)
- \( b \) maps to \( t \) (1 image)
- \( c \) maps to \( s \) (1 image)
- \( d \) maps to \( u \) (1 image)

However, there is a mistake in the transcript regarding the mapping for \( h \). It states that \( h(a) = u \) is also included, which means:
- \( a \) has two images: \( s \) and \( u \).

Since \( a \) maps to two different images, \( h \) does not satisfy the definition of a function.

### Step 4: Conclusion
Based on our analysis:
- \( f \) is a function.
- \( h \) is not a function.

### Final Answer
The correct statement is:
**Option B: \( f \) is a function and \( h \) is not a function.**