Let .G; / be an Abelian group. Define the following two subsets of G: a. H D fx 2 G W x x D eg, and b. K D fx 2 G W x D y y for some y 2 Gg. Prove that .H; / and .K; / are subgroups of .G; /. Furthermore, give examples to demonstrate that if the requirement that .G; / be Abelian is deleted, H and K do not necessarily constitute subgroups. 7. Let

MTH 111 (Study Skills #2) Homework – Why – your brain is like a muscle – studying and doing homework literally builds “connections” in your brain – brain strength comes from connections your brain builds when you study or practice – These connections are necessary for your brain’s ability to perform what you want it todo (such as solving equations – your brain learns by doing! – WATCHING someone solve math problems trains your brain to get really good at WATCHING someone solve math problems – If YOU actively solve mat problems yourself, YOU grow connections for solving math problems – Practice makes your connections more awesome – When you practice: – your connections become stronger (and thus information passes through them more easily – your connections become faster – makes brain work faster – your brain becomes more efﬁcient and more capable of processing MORE NEW information – Your short term memory is very short! – if you learn something new and do it only once or twice, the connection that is created is very weak and can disappear within hours – within 20 minutes, you remember only 60% – within 24 hours, you remember only 30% – but, if you practice within 24 hours then practice again later, you remember 80% – this is why instructors stre