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Answer :
Certainly! Let's determine which of the provided sets is a subset of the set [tex]\( A \)[/tex].
First, recall that a set [tex]\( B \)[/tex] is a subset of the set [tex]\( A \)[/tex] if every element in [tex]\( B \)[/tex] is also in [tex]\( A \)[/tex].
Given:
[tex]\[ A = \{14, 24, 46, 94, 99, 89, 78\} \][/tex]
We need to check which of the following sets is a subset of [tex]\( A \)[/tex]:
1. [tex]\( D = \{14, 24, 78, 46, 99, 94\} \)[/tex]
2. [tex]\( E = \{14, 46, 94, 89, 78, 91, 99\} \)[/tex]
3. [tex]\( F = \{89, 99, 49, 46, 78, 14\} \)[/tex]
4. [tex]\( C = \{78, 14, 64, 99, 89, 46, 94, 24\} \)[/tex]
### Checking Set [tex]\( D \)[/tex]
Elements in [tex]\( D \)[/tex] are [tex]\( 14, 24, 78, 46, 99, 94 \)[/tex].
All these elements are in [tex]\( A \)[/tex]:
[tex]\[ 14 \in A, 24 \in A, 78 \in A, 46 \in A, 99 \in A, 94 \in A \][/tex]
So, [tex]\( D \)[/tex] is a subset of [tex]\( A \)[/tex].
### Checking Set [tex]\( E \)[/tex]
Elements in [tex]\( E \)[/tex] are [tex]\( 14, 46, 94, 89, 78, 91, 99 \)[/tex].
In this case, [tex]\( 91 \notin A \)[/tex].
Since [tex]\( 91 \)[/tex] is not in [tex]\( A \)[/tex], [tex]\( E \)[/tex] is not a subset of [tex]\( A \)[/tex].
### Checking Set [tex]\( F \)[/tex]
Elements in [tex]\( F \)[/tex] are [tex]\( 89, 99, 49, 46, 78, 14 \)[/tex].
Checking these:
[tex]\[ 49 \notin A \][/tex]
Since [tex]\( 49 \)[/tex] is not in [tex]\( A \)[/tex], [tex]\( F \)[/tex] is not a subset of [tex]\( A \)[/tex].
### Checking Set [tex]\( C \)[/tex]
Elements in [tex]\( C \)[/tex] are [tex]\( 78, 14, 64, 99, 89, 46, 94, 24 \)[/tex].
Checking these:
[tex]\[ 64 \notin A \][/tex]
Since [tex]\( 64 \)[/tex] is not in [tex]\( A \)[/tex], [tex]\( C \)[/tex] is not a subset of [tex]\( A \)[/tex].
### Conclusion
Given our checks, only set [tex]\( D = \{14, 24, 78, 46, 99, 94\} \)[/tex] is a subset of [tex]\( A \)[/tex]. Thus, the answer is:
[tex]\[ \{D\} \][/tex]
First, recall that a set [tex]\( B \)[/tex] is a subset of the set [tex]\( A \)[/tex] if every element in [tex]\( B \)[/tex] is also in [tex]\( A \)[/tex].
Given:
[tex]\[ A = \{14, 24, 46, 94, 99, 89, 78\} \][/tex]
We need to check which of the following sets is a subset of [tex]\( A \)[/tex]:
1. [tex]\( D = \{14, 24, 78, 46, 99, 94\} \)[/tex]
2. [tex]\( E = \{14, 46, 94, 89, 78, 91, 99\} \)[/tex]
3. [tex]\( F = \{89, 99, 49, 46, 78, 14\} \)[/tex]
4. [tex]\( C = \{78, 14, 64, 99, 89, 46, 94, 24\} \)[/tex]
### Checking Set [tex]\( D \)[/tex]
Elements in [tex]\( D \)[/tex] are [tex]\( 14, 24, 78, 46, 99, 94 \)[/tex].
All these elements are in [tex]\( A \)[/tex]:
[tex]\[ 14 \in A, 24 \in A, 78 \in A, 46 \in A, 99 \in A, 94 \in A \][/tex]
So, [tex]\( D \)[/tex] is a subset of [tex]\( A \)[/tex].
### Checking Set [tex]\( E \)[/tex]
Elements in [tex]\( E \)[/tex] are [tex]\( 14, 46, 94, 89, 78, 91, 99 \)[/tex].
In this case, [tex]\( 91 \notin A \)[/tex].
Since [tex]\( 91 \)[/tex] is not in [tex]\( A \)[/tex], [tex]\( E \)[/tex] is not a subset of [tex]\( A \)[/tex].
### Checking Set [tex]\( F \)[/tex]
Elements in [tex]\( F \)[/tex] are [tex]\( 89, 99, 49, 46, 78, 14 \)[/tex].
Checking these:
[tex]\[ 49 \notin A \][/tex]
Since [tex]\( 49 \)[/tex] is not in [tex]\( A \)[/tex], [tex]\( F \)[/tex] is not a subset of [tex]\( A \)[/tex].
### Checking Set [tex]\( C \)[/tex]
Elements in [tex]\( C \)[/tex] are [tex]\( 78, 14, 64, 99, 89, 46, 94, 24 \)[/tex].
Checking these:
[tex]\[ 64 \notin A \][/tex]
Since [tex]\( 64 \)[/tex] is not in [tex]\( A \)[/tex], [tex]\( C \)[/tex] is not a subset of [tex]\( A \)[/tex].
### Conclusion
Given our checks, only set [tex]\( D = \{14, 24, 78, 46, 99, 94\} \)[/tex] is a subset of [tex]\( A \)[/tex]. Thus, the answer is:
[tex]\[ \{D\} \][/tex]
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