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Which of the following is a subset of [tex] S = \{\text{dog, cat, pig, rabbit, squirrel, fish, hamster, elephant, zebra, snail}\} [/tex]?

A. [tex] J = \{\text{rabbit, squirrel, hamster, zebra}\} [/tex]
B. [tex] K = \{\text{dog, cat, duck, snail}\} [/tex]
C. [tex] L = \{\text{dog, cat, pig, hyena}\} [/tex]
D. [tex] M = \{\text{rabbit, squirrel, hamster, snake}\} [/tex]


Sagot :

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].