### GR1268 Q12, 23, 29

Posted:

**Wed Oct 26, 2016 11:41 am**Hi! I come from a physics background, so I wanted to check my understanding of some of the terminology around set theory questions. Please let me know if there are any mistakes!

Q12 asks which integers n within $$3 \le n \le 11$$ has only one group of order n up to isomorphism. I understand this as "How many sets with n elements exist that cannot be mapped onto each other?"

Q23 begins "Let $$\left ( \mathbb{Z}_{10}, +, \cdot \right )$$ be the ring of integers modulo 10, and let S be the subset of $$\mathbb{Z}_{10}$$ represented by {0,2,4,6,8}." My understanding is that $$\left ( \mathbb{Z}_{10}, +, \cdot \right )$$ is just notation that they are introducing for this problem, corresponding to the integers up to 10 modulo 10, which is a ring because that's how modulo works. The question goes on to ask about whether the introduced operator applied to S is closed under addition modulo 10. This means that adding any elements of the set and then taking the modulo will yield a member of the set.

Q29 defines a tree as a connected graph with no cycles. I'm actually really confused about what this means. Does it just mean that the graph cannot loop back onto itself? And by extension that differently angled positions of the vertices are considered the same tree?

Q12 asks which integers n within $$3 \le n \le 11$$ has only one group of order n up to isomorphism. I understand this as "How many sets with n elements exist that cannot be mapped onto each other?"

Q23 begins "Let $$\left ( \mathbb{Z}_{10}, +, \cdot \right )$$ be the ring of integers modulo 10, and let S be the subset of $$\mathbb{Z}_{10}$$ represented by {0,2,4,6,8}." My understanding is that $$\left ( \mathbb{Z}_{10}, +, \cdot \right )$$ is just notation that they are introducing for this problem, corresponding to the integers up to 10 modulo 10, which is a ring because that's how modulo works. The question goes on to ask about whether the introduced operator applied to S is closed under addition modulo 10. This means that adding any elements of the set and then taking the modulo will yield a member of the set.

Q29 defines a tree as a connected graph with no cycles. I'm actually really confused about what this means. Does it just mean that the graph cannot loop back onto itself? And by extension that differently angled positions of the vertices are considered the same tree?