Answer :
Let's define the given sets and constraints step by step:
1. The universal set \( U \) is the set of all real numbers, i.e., \( U = \{ x \mid x \text{ is a real number} \} \).
2. Set \( A \) is defined as \( A = \{ x \mid x \in U \text{ and } x + 2 > 10 \} \). Simplifying this condition:
[tex]\[ x + 2 > 10 \][/tex]
[tex]\[ x > 8 \][/tex]
So \( A = \{ x \mid x > 8 \} \).
3. Set \( B \) is defined as \( B = \{ x \mid x \in U \text{ and } 2x > 10 \} \). Simplifying this condition:
[tex]\[ 2x > 10 \][/tex]
[tex]\[ x > 5 \][/tex]
So \( B = \{ x \mid x > 5 \} \).
Now, let's evaluate each pair of statements given in the problem:
Statement 1: \(5 \notin A ; 5 \in B\)
- For \( 5 \notin A \):
[tex]\[ 5 \notin A \text{ is true because } 5 \leq 8. \][/tex]
- For \( 5 \in B \):
[tex]\[ 5 \in B \text{ is false because } 5 \leq 5. \][/tex]
These simplify to:
[tex]\[ (1, 0) \][/tex]
Statement 2: \(6 \in A ; 6 \notin B\)
- For \( 6 \in A \):
[tex]\[ 6 \in A \text{ is false because } 6 \leq 8. \][/tex]
- For \( 6 \notin B \):
[tex]\[ 6 \notin B \text{ is false because } 6 > 5. \][/tex]
These simplify to:
[tex]\[ (0, 0) \][/tex]
Statement 3: \(8 \notin A ; 8 \in B\)
- For \( 8 \notin A \):
[tex]\[ 8 \notin A \text{ is true because } 8 \leq 8. \][/tex]
- For \( 8 \in B \):
[tex]\[ 8 \in B \text{ is true because } 8 > 5. \][/tex]
These simplify to:
[tex]\[ (1, 1) \][/tex]
Statement 4: \(9 \in A ; 9 \notin B\)
- For \( 9 \in A \):
[tex]\[ 9 \in A \text{ is true because } 9 > 8. \][/tex]
- For \( 9 \notin B \):
[tex]\[ 9 \notin B \text{ is false because } 9 > 5. \][/tex]
These simplify to:
[tex]\[ (1, 0) \][/tex]
Thus, the correct pairs of statements corresponding to the given conditions are:
[tex]\[ ((1, 0), (0, 0), (1, 1), (1, 0)) \][/tex]
1. The universal set \( U \) is the set of all real numbers, i.e., \( U = \{ x \mid x \text{ is a real number} \} \).
2. Set \( A \) is defined as \( A = \{ x \mid x \in U \text{ and } x + 2 > 10 \} \). Simplifying this condition:
[tex]\[ x + 2 > 10 \][/tex]
[tex]\[ x > 8 \][/tex]
So \( A = \{ x \mid x > 8 \} \).
3. Set \( B \) is defined as \( B = \{ x \mid x \in U \text{ and } 2x > 10 \} \). Simplifying this condition:
[tex]\[ 2x > 10 \][/tex]
[tex]\[ x > 5 \][/tex]
So \( B = \{ x \mid x > 5 \} \).
Now, let's evaluate each pair of statements given in the problem:
Statement 1: \(5 \notin A ; 5 \in B\)
- For \( 5 \notin A \):
[tex]\[ 5 \notin A \text{ is true because } 5 \leq 8. \][/tex]
- For \( 5 \in B \):
[tex]\[ 5 \in B \text{ is false because } 5 \leq 5. \][/tex]
These simplify to:
[tex]\[ (1, 0) \][/tex]
Statement 2: \(6 \in A ; 6 \notin B\)
- For \( 6 \in A \):
[tex]\[ 6 \in A \text{ is false because } 6 \leq 8. \][/tex]
- For \( 6 \notin B \):
[tex]\[ 6 \notin B \text{ is false because } 6 > 5. \][/tex]
These simplify to:
[tex]\[ (0, 0) \][/tex]
Statement 3: \(8 \notin A ; 8 \in B\)
- For \( 8 \notin A \):
[tex]\[ 8 \notin A \text{ is true because } 8 \leq 8. \][/tex]
- For \( 8 \in B \):
[tex]\[ 8 \in B \text{ is true because } 8 > 5. \][/tex]
These simplify to:
[tex]\[ (1, 1) \][/tex]
Statement 4: \(9 \in A ; 9 \notin B\)
- For \( 9 \in A \):
[tex]\[ 9 \in A \text{ is true because } 9 > 8. \][/tex]
- For \( 9 \notin B \):
[tex]\[ 9 \notin B \text{ is false because } 9 > 5. \][/tex]
These simplify to:
[tex]\[ (1, 0) \][/tex]
Thus, the correct pairs of statements corresponding to the given conditions are:
[tex]\[ ((1, 0), (0, 0), (1, 1), (1, 0)) \][/tex]
