Question

Fine the indicated set if
A={x|x≥-2}
B={x|x<4}
C={x|-1<1x≤5}
(a) BuC (b) BnC (c) AnB (d) AnC

Answer 1

Given the sets:

- \( A = \{ x \mid x \geq -2 \} \)
- \( B = \{ x \mid x < 4 \} \)
- \( C = \{ x \mid -1 < x \leq 5 \} \)

Let's find the indicated sets:

### (a) \( B \cup C \)

The union \( B \cup C \) consists of all elements that are in either set \( B \) or set \( C \).

- \( B = (-\infty, 4) \) (all numbers less than 4)
- \( C = (-1, 5] \) (numbers greater than -1 and less than or equal to 5)

To find \( B \cup C \):
- The range of \( B \) goes to 4 and the range of \( C \) goes to 5.
- Therefore, \( B \cup C = (-\infty, 5] \).

### (b) \( B \cap C \)

The intersection \( B \cap C \) consists of all elements that are in both set \( B \) and set \( C \).

We find the overlap:
- For \( B = (-\infty, 4) \)
- For \( C = (-1, 5] \)

To find \( B \cap C \), we need the range where both conditions are satisfied:
- The lower bound is \( -1 \) (from \( C \)), and the upper bound is \( 4 \) (from \( B \)).
- Thus, \( B \cap C = (-1, 4) \).

### (c) \( A \cap B \)

The intersection \( A \cap B \) consists of all elements that are in both set \( A \) and set \( B \).

- \( A = [-2, \infty) \)
- \( B = (-\infty, 4) \)

To find \( A \cap B \):
- The lower bound is \( -2 \), and the upper bound is \( 4 \) (but does not include 4 because it’s the upper limit of \( B \)).
- Thus, \( A \cap B = [-2, 4) \).

### (d) \( A \cap C \)

The intersection \( A \cap C \) consists of all elements that are in both set \( A \) and set \( C \).

- \( A = [-2, \infty) \)
- \( C = (-1, 5] \)

To find \( A \cap C \):
- The lower bound is \( -1 \), and the upper bound is \( 5 \) (including 5 because it’s part of \( C \)).
- Hence, \( A \cap C = (-1, 5] \).

### Summary of Results:

- (a) \( B \cup C = (-\infty, 5] \)
- (b) \( B \cap C = (-1, 4) \)
- (c) \( A \cap B = [-2, 4) \)
- (d) \( A \cap C = (-1, 5] \)