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

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) $\{n \in \mathbb{Z}: n \geq 0\}$.