Consider the following statements : `S_(1)` : If `A={a}` and `B = {a, b, c},` then `A in B`. `S_(2)` : If `n(A)=x,` then `n(P(A))=2^(x)`. `S_(3)` : `(AuuB)sube A`. `S_(4) : phi` is subset of every set State, in order, whether `S_(1), S_(2), S_(3), S_(4)` and true or false (a)`T T T T` (b)`FT T F` (c)`FTFT` (d)`FFT T`

Options

A. `T T T T`
B. `FT T F`
C. `FTFT`
D. `FFT T`

Explanation

To solve the question, we need to evaluate each statement \( S_1, S_2, S_3, \) and \( S_4 \) and determine whether they are true or false.

### Step-by-step Solution:

1. **Evaluate Statement \( S_1 \)**:
- Statement: If \( A = \{a\} \) and \( B = \{a, b, c\} \), then \( A \in B \).
- Analysis: Here, \( A \) is a set containing the element \( a \), while \( B \) is a set containing the elements \( a, b, \) and \( c \). The statement claims that \( A \) (which is a set) is an element of \( B \). However, \( A \) is not an element of \( B \); rather, \( a \) is an element of \( B \). Therefore, this statement is **False**.

2. **Evaluate Statement \( S_2 \)**:
- Statement: If \( n(A) = x \), then \( n(P(A)) = 2^x \).
- Analysis: The power set \( P(A) \) of a set \( A \) contains all possible subsets of \( A \). If \( A \) has \( x \) elements, then the number of subsets (including the empty set and \( A \) itself) is \( 2^x \). Thus, this statement is **True**.

3. **Evaluate Statement \( S_3 \)**:
- Statement: \( A \cup B \subseteq A \).
- Analysis: The union \( A \cup B \) contains all elements that are in either \( A \) or \( B \). For example, if \( A = \{1, 2\} \) and \( B = \{3, 4\} \), then \( A \cup B = \{1, 2, 3, 4\} \), which is not a subset of \( A \). Hence, this statement is **False**.

4. **Evaluate Statement \( S_4 \)**:
- Statement: \( \phi \) is a subset of every set.
- Analysis: The empty set \( \phi \) is indeed a subset of every set by definition. Therefore, this statement is **True**.

### Summary of Statements:
- \( S_1 \): False
- \( S_2 \): True
- \( S_3 \): False
- \( S_4 \): True

### Final Result:
The order of truth values for the statements is: **FTFT**.

### Answer:
(c) FTFT