Answer :
Sure, I'll provide a step-by-step solution for each part of the question.
Given:
[tex]\[ \begin{array}{l} \delta = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \\ A = \{2, 3, 5, 7\} \\ B = \{4, 6, 8, 10\} \end{array} \][/tex]
### (a) Explain why [tex]\( A \cap B = \varnothing \)[/tex]
To understand why [tex]\( A \cap B = \varnothing \)[/tex], we need to examine the elements of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- Set [tex]\( A \)[/tex] consists of elements: [tex]\( \{2, 3, 5, 7\} \)[/tex]
- Set [tex]\( B \)[/tex] consists of elements: [tex]\( \{4, 6, 8, 10\} \)[/tex]
The intersection [tex]\( A \cap B \)[/tex] represents the elements that are common to both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. Since there are no common elements between [tex]\( A \)[/tex] and [tex]\( B \)[/tex], their intersection is an empty set.
[tex]\[ A \cap B = \varnothing \][/tex]
So, the explanation is:
"Because there are no common elements in [tex]\( A \)[/tex] and [tex]\( B \)[/tex]."
### (b) Write down the two possible values of [tex]\( x \)[/tex]
We are given that [tex]\( x \)[/tex] is in the universal set [tex]\( \delta \)[/tex] but not in the union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
First, find the union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A \cup B = \{2, 3, 4, 5, 6, 7, 8, 10\} \][/tex]
Next, identify the elements in [tex]\( \delta \)[/tex] that are not in [tex]\( A \cup B \)[/tex]:
[tex]\[ \delta - (A \cup B) = \{1, 9\} \][/tex]
Therefore, the two possible values of [tex]\( x \)[/tex] are [tex]\( 1 \)[/tex] and [tex]\( 9 \)[/tex].
### (c) List all the members of set [tex]\( C \)[/tex]
We are given the following conditions for set [tex]\( C \)[/tex]:
[tex]\[ \begin{array}{l} A \cup B \cup C = \delta \\ A \cap C = \{2\} \\ B \cap C' = \{4, 6, 10\} \end{array} \][/tex]
To satisfy these conditions, let's determine the elements of [tex]\( C \)[/tex]:
1. Since [tex]\( A \cup B \cup C = \delta \)[/tex], any element in [tex]\( \delta \)[/tex] must be in [tex]\( A \)[/tex], [tex]\( B \)[/tex], or [tex]\( C \)[/tex].
2. [tex]\( A \cap C = \{2\} \)[/tex] means that [tex]\( 2 \)[/tex] is the only element that is common to both [tex]\( A \)[/tex] and [tex]\( C \)[/tex].
3. [tex]\( B \cap C' = \{4, 6, 10\} \)[/tex] indicates that [tex]\( 4 \)[/tex], [tex]\( 6 \)[/tex], and [tex]\( 10 \)[/tex] are in set [tex]\( B \)[/tex] but not in set [tex]\( C \)[/tex].
Let's start by constructing set [tex]\( C \)[/tex]:
- Since [tex]\( 4 \)[/tex], [tex]\( 6 \)[/tex], [tex]\( 8 \)[/tex], and [tex]\( 10 \)[/tex] must be excluded from [tex]\( C \)[/tex], they are in [tex]\( B \)[/tex].
- To ensure [tex]\( A \cap C = \{2\} \)[/tex], include [tex]\( 2 \)[/tex] in [tex]\( C \)[/tex].
- The remaining numbers [tex]\( 1 \)[/tex] and [tex]\( 9 \)[/tex] (which are neither in [tex]\( A \)[/tex] nor in [tex]\( B \)[/tex]) should be in [tex]\( C \)[/tex] to satisfy [tex]\( A \cup B \cup C = \delta \)[/tex].
Therefore, the members of set [tex]\( C \)[/tex] are:
[tex]\[ C = \{1, 2, 9\} \][/tex]
### Summary
(a) Because there are no common elements in [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
(b) The two possible values of [tex]\( x \)[/tex] are [tex]\( 1 \)[/tex] and [tex]\( 9 \)[/tex].
(c) The members of set [tex]\( C \)[/tex] are [tex]\( \{1, 2, 9\} \)[/tex].
Given:
[tex]\[ \begin{array}{l} \delta = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \\ A = \{2, 3, 5, 7\} \\ B = \{4, 6, 8, 10\} \end{array} \][/tex]
### (a) Explain why [tex]\( A \cap B = \varnothing \)[/tex]
To understand why [tex]\( A \cap B = \varnothing \)[/tex], we need to examine the elements of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
- Set [tex]\( A \)[/tex] consists of elements: [tex]\( \{2, 3, 5, 7\} \)[/tex]
- Set [tex]\( B \)[/tex] consists of elements: [tex]\( \{4, 6, 8, 10\} \)[/tex]
The intersection [tex]\( A \cap B \)[/tex] represents the elements that are common to both sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]. Since there are no common elements between [tex]\( A \)[/tex] and [tex]\( B \)[/tex], their intersection is an empty set.
[tex]\[ A \cap B = \varnothing \][/tex]
So, the explanation is:
"Because there are no common elements in [tex]\( A \)[/tex] and [tex]\( B \)[/tex]."
### (b) Write down the two possible values of [tex]\( x \)[/tex]
We are given that [tex]\( x \)[/tex] is in the universal set [tex]\( \delta \)[/tex] but not in the union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
First, find the union of sets [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ A \cup B = \{2, 3, 4, 5, 6, 7, 8, 10\} \][/tex]
Next, identify the elements in [tex]\( \delta \)[/tex] that are not in [tex]\( A \cup B \)[/tex]:
[tex]\[ \delta - (A \cup B) = \{1, 9\} \][/tex]
Therefore, the two possible values of [tex]\( x \)[/tex] are [tex]\( 1 \)[/tex] and [tex]\( 9 \)[/tex].
### (c) List all the members of set [tex]\( C \)[/tex]
We are given the following conditions for set [tex]\( C \)[/tex]:
[tex]\[ \begin{array}{l} A \cup B \cup C = \delta \\ A \cap C = \{2\} \\ B \cap C' = \{4, 6, 10\} \end{array} \][/tex]
To satisfy these conditions, let's determine the elements of [tex]\( C \)[/tex]:
1. Since [tex]\( A \cup B \cup C = \delta \)[/tex], any element in [tex]\( \delta \)[/tex] must be in [tex]\( A \)[/tex], [tex]\( B \)[/tex], or [tex]\( C \)[/tex].
2. [tex]\( A \cap C = \{2\} \)[/tex] means that [tex]\( 2 \)[/tex] is the only element that is common to both [tex]\( A \)[/tex] and [tex]\( C \)[/tex].
3. [tex]\( B \cap C' = \{4, 6, 10\} \)[/tex] indicates that [tex]\( 4 \)[/tex], [tex]\( 6 \)[/tex], and [tex]\( 10 \)[/tex] are in set [tex]\( B \)[/tex] but not in set [tex]\( C \)[/tex].
Let's start by constructing set [tex]\( C \)[/tex]:
- Since [tex]\( 4 \)[/tex], [tex]\( 6 \)[/tex], [tex]\( 8 \)[/tex], and [tex]\( 10 \)[/tex] must be excluded from [tex]\( C \)[/tex], they are in [tex]\( B \)[/tex].
- To ensure [tex]\( A \cap C = \{2\} \)[/tex], include [tex]\( 2 \)[/tex] in [tex]\( C \)[/tex].
- The remaining numbers [tex]\( 1 \)[/tex] and [tex]\( 9 \)[/tex] (which are neither in [tex]\( A \)[/tex] nor in [tex]\( B \)[/tex]) should be in [tex]\( C \)[/tex] to satisfy [tex]\( A \cup B \cup C = \delta \)[/tex].
Therefore, the members of set [tex]\( C \)[/tex] are:
[tex]\[ C = \{1, 2, 9\} \][/tex]
### Summary
(a) Because there are no common elements in [tex]\( A \)[/tex] and [tex]\( B \)[/tex].
(b) The two possible values of [tex]\( x \)[/tex] are [tex]\( 1 \)[/tex] and [tex]\( 9 \)[/tex].
(c) The members of set [tex]\( C \)[/tex] are [tex]\( \{1, 2, 9\} \)[/tex].
