Pergunta

13. Let Z be the group of all integers under the operation of addition. Which of the following subsets of Z is NOT a subgroup of Z? (A) 0 (B) nin Z:ngeqslant 0 c nin Z:nis an even integer (D) nin Z:nis divisible by both6and9
Solução

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ZefaMestre · Tutor por 5 anos
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To determine which subset is not a subgroup of \mathbb{Z} , we need to check the subgroup criteria: closure under addition, existence of an identity element, and existence of inverses.
- (A) \{0\} : This is a subgroup because it contains the identity element 0, and is closed under addition (0 + 0 = 0).
- (B) \{n \in \mathbb{Z}: n \geq 0\} : This set is not a subgroup because it does not contain inverses for all its elements. For example, if n = 1 , then -1 is not in the set.
- (C) \{n \in \mathbb{Z}: n is an even integer \} : This is a subgroup because the sum of any two even integers is even, and the inverse of an even integer is also even.
- (D) \{n \in \mathbb{Z}: n is divisible by both 6 and 9 \} : This is a subgroup because it consists of integers divisible by the least common multiple of 6 and 9, which is 18. It is closed under addition and contains inverses.
- (E) \mathbb{Z} : The group itself is always a subgroup.
Therefore, the subset that is NOT a subgroup of \mathbb{Z} is (B) \{n \in \mathbb{Z}: n \geq 0\} .
- (A) \{0\} : This is a subgroup because it contains the identity element 0, and is closed under addition (0 + 0 = 0).
- (B) \{n \in \mathbb{Z}: n \geq 0\} : This set is not a subgroup because it does not contain inverses for all its elements. For example, if n = 1 , then -1 is not in the set.
- (C) \{n \in \mathbb{Z}: n is an even integer \} : This is a subgroup because the sum of any two even integers is even, and the inverse of an even integer is also even.
- (D) \{n \in \mathbb{Z}: n is divisible by both 6 and 9 \} : This is a subgroup because it consists of integers divisible by the least common multiple of 6 and 9, which is 18. It is closed under addition and contains inverses.
- (E) \mathbb{Z} : The group itself is always a subgroup.
Therefore, the subset that is NOT a subgroup of \mathbb{Z} is (B) \{n \in \mathbb{Z}: n \geq 0\} .
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