To determine which of the given sets [tex]\( J \)[/tex], [tex]\( K \)[/tex], [tex]\( L \)[/tex], and [tex]\( M \)[/tex] are subsets of the set [tex]\( S \)[/tex], we need to check whether every element of these sets is also an element of [tex]\( S \)[/tex].
Given:
[tex]\[ S = \{ \text{dog}, \text{cat}, \text{pig}, \text{rabbit}, \text{squirrel}, \text{fish}, \text{hamster}, \text{elephant}, \text{zebra}, \text{snail} \} \][/tex]
Let's check each set one by one:
1. Set [tex]\( J \)[/tex]:
[tex]\[ J = \{ \text{rabbit}, \text{squirrel}, \text{hamster}, \text{zebra} \} \][/tex]
Checking elements of [tex]\( J \)[/tex] in [tex]\( S \)[/tex]:
- "rabbit" is in [tex]\( S \)[/tex]
- "squirrel" is in [tex]\( S \)[/tex]
- "hamster" is in [tex]\( S \)[/tex]
- "zebra" is in [tex]\( S \)[/tex]
Since all elements of [tex]\( J \)[/tex] are in [tex]\( S \)[/tex], [tex]\( J \)[/tex] is a subset of [tex]\( S \)[/tex]:
[tex]\[ J \subseteq S \][/tex]
2. Set [tex]\( K \)[/tex]:
[tex]\[ K = \{ \text{dog}, \text{cat}, \text{duck}, \text{snail} \} \][/tex]
Checking elements of [tex]\( K \)[/tex] in [tex]\( S \)[/tex]:
- "dog" is in [tex]\( S \)[/tex]
- "cat" is in [tex]\( S \)[/tex]
- "duck" is not in [tex]\( S \)[/tex]
- "snail" is in [tex]\( S \)[/tex]
Since "duck" is not in [tex]\( S \)[/tex], [tex]\( K \)[/tex] is not a subset of [tex]\( S \)[/tex]:
[tex]\[ K \nsubseteq S \][/tex]
3. Set [tex]\( L \)[/tex]:
[tex]\[ L = \{ \text{dog}, \text{cat}, \text{pig}, \text{hyena} \} \][/tex]
Checking elements of [tex]\( L \)[/tex] in [tex]\( S \)[/tex]:
- "dog" is in [tex]\( S \)[/tex]
- "cat" is in [tex]\( S \)[/tex]
- "pig" is in [tex]\( S \)[/tex]
- "hyena" is not in [tex]\( S \)[/tex]
Since "hyena" is not in [tex]\( S \)[/tex], [tex]\( L \)[/tex] is not a subset of [tex]\( S \)[/tex]:
[tex]\[ L \nsubseteq S \][/tex]
4. Set [tex]\( M \)[/tex]:
[tex]\[ M = \{ \text{rabbit}, \text{squirrel}, \text{hamster}, \text{snake} \} \][/tex]
Checking elements of [tex]\( M \)[/tex] in [tex]\( S \)[/tex]:
- "rabbit" is in [tex]\( S \)[/tex]
- "squirrel" is in [tex]\( S \)[/tex]
- "hamster" is in [tex]\( S \)[/tex]
- "snake" is not in [tex]\( S \)[/tex]
Since "snake" is not in [tex]\( S \)[/tex], [tex]\( M \)[/tex] is not a subset of [tex]\( S \)[/tex]:
[tex]\[ M \nsubseteq S \][/tex]
Based on the above analysis, the results are:
- [tex]\( J \)[/tex] is a subset of [tex]\( S \)[/tex]: [tex]\[ J \subseteq S \][/tex]
- [tex]\( K \)[/tex] is not a subset of [tex]\( S \)[/tex]
- [tex]\( L \)[/tex] is not a subset of [tex]\( S \)[/tex]
- [tex]\( M \)[/tex] is not a subset of [tex]\( S \)[/tex]
Thus, the only subset of [tex]\( S \)[/tex] from the given sets is [tex]\( J \)[/tex].
