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(Socks–Shoes Property) Draw an analogy between the
Chapter 2, Problem 24E(choose chapter or problem)
(Socks–Shoes Property) Draw an analogy between the statement (ab)–1 = b–1 a–1 and the act of putting on and taking off your socks and shoes. Find distinct nonidentity elements a and b from a non-Abelian group such that (ab)–1 = a–1 b–1. Find an example that shows that in a group, it is possible to have (ab)–2 ? b–2 a–2. What would be an appropriate name for the group property (abc)–1 = c–1 b–1 a–1?
Questions & Answers
QUESTION:
(Socks–Shoes Property) Draw an analogy between the statement (ab)–1 = b–1 a–1 and the act of putting on and taking off your socks and shoes. Find distinct nonidentity elements a and b from a non-Abelian group such that (ab)–1 = a–1 b–1. Find an example that shows that in a group, it is possible to have (ab)–2 ? b–2 a–2. What would be an appropriate name for the group property (abc)–1 = c–1 b–1 a–1?
ANSWER:Step 1 of 4
Sock- Shoe Property: This property is a direct analogy from real life to mathematical manipulations. If one thinks of as putting on socks, as putting on shoes, , as taking off socks, and as taking off shoes, the theorem demonstrates the order in which one must perform these actions.
So, here represents the action as: Putting on socks followed by putting on shoes.
So, the term represents the action as: Putting off shoes followed by putting off socks.
Also, the term represents the action as: Putting off socks followed by putting off shoes.
But you cannot put off socks before putting off shoes. So, for Sock-Shoe property is not true. So, .