Let A = {p,q,r,s},B={p,q,r} and C= {q,S} . Find all sets X such that :
(i) `XsubBandXsubC`
(ii) `XsubAandX!inB`.

Answer

To solve the question, we need to find all sets \( X \) that satisfy the given conditions. We will address each part of the question step by step.

### Given Sets:
- \( A = \{p, q, r, s\} \)
- \( B = \{p, q, r\} \)
- \( C = \{q, s\} \)

### Part (i): Find all sets \( X \) such that \( X \subseteq B \) and \( X \subseteq C \).

1. **Identify subsets of set B**:
- The subsets of \( B \) are:
- \( \emptyset \) (the empty set)
- \( \{p\} \)
- \( \{q\} \)
- \( \{r\} \)
- \( \{p, q\} \)
- \( \{p, r\} \)
- \( \{q, r\} \)
- \( \{p, q, r\} \)

2. **Identify subsets of set C**:
- The subsets of \( C \) are:
- \( \emptyset \)
- \( \{q\} \)
- \( \{s\} \)
- \( \{q, s\} \)

3. **Find common subsets**:
- We need to find subsets that are common to both \( B \) and \( C \).
- The only common element between \( B \) and \( C \) is \( q \).
- Therefore, the possible sets \( X \) that are subsets of both \( B \) and \( C \) are:
- \( \emptyset \)
- \( \{q\} \)

### Conclusion for Part (i):
The sets \( X \) that satisfy \( X \subseteq B \) and \( X \subseteq C \) are:
- \( X = \emptyset \)
- \( X = \{q\} \)

### Part (ii): Find all sets \( X \) such that \( X \subseteq A \) and \( X \not\subseteq B \).

1. **Identify subsets of set A**:
- The subsets of \( A \) are:
- \( \emptyset \)
- \( \{p\} \)
- \( \{q\} \)
- \( \{r\} \)
- \( \{s\} \)
- \( \{p, q\} \)
- \( \{p, r\} \)
- \( \{q, r\} \)
- \( \{p, s\} \)
- \( \{q, s\} \)
- \( \{r, s\} \)
- \( \{p, q, r\} \)
- \( \{p, q, s\} \)
- \( \{p, r, s\} \)
- \( \{q, r, s\} \)
- \( \{p, q, r, s\} \)

2. **Determine which subsets are not in B**:
- The elements of \( B \) are \( \{p, q, r\} \).
- Therefore, any subset of \( A \) that contains \( s \) will not be a subset of \( B \).
- The valid subsets of \( A \) that contain \( s \) are:
- \( \{s\} \)
- \( \{q, s\} \)
- \( \{p, s\} \)
- \( \{p, q, s\} \)
- \( \{r, s\} \)
- \( \{p, r, s\} \)
- \( \{q, r, s\} \)
- \( \{p, q, r, s\} \)

### Conclusion for Part (ii):
The sets \( X \) that satisfy \( X \subseteq A \) and \( X \not\subseteq B \) are:
- \( X = \{s\} \)
- \( X = \{q, s\} \)
- \( X = \{p, s\} \)
- \( X = \{p, q, s\} \)
- \( X = \{r, s\} \)
- \( X = \{p, r, s\} \)
- \( X = \{q, r, s\} \)
- \( X = \{p, q, r, s\} \)

### Final Answer:
1. For part (i): \( X = \emptyset \) or \( X = \{q\} \)
2. For part (ii): \( X = \{s\}, \{q, s\}, \{p, s\}, \{p, q, s\}, \{r, s\}, \{p, r, s\}, \{q, r, s\}, \{p, q, r, s\} \)