Discrete Math Problems

Question

1. State the principle of Mathematical Induction.
2. Prove by mathematical induction that n² is the sum of the first n odd numbers.
3. State the well-ordering principle for the set of natural numbers.
4. Deduce the principle of induction from the well-ordering principle.
5. Define that(G,∗,⁻¹,e) is a group.
6. (a) Describe the additive group of integers modulo n. You have to state exactly what its elements are, how addition is defined, what the identity is, and what the additive inverse of an element is.
(b) Find the additive inverse of 40 modulo 48.
(c) Solve15+x=7modulo48.
7. Integers modulo 24 lead to what is called clock arithmetic. For example, if it is now 11−o’clock than 28 hours later it will be 11 + 28 = 11 + 4 = 15−o’clock (which is 3pm, of course a day later). Assume that a flight from Houston to Beijing leaves at 12−o’clock (noon). The flight lasts 15 hours. What will be the time in Houston at arrival in Beijing, and what will be the local time in Beijing? Notice that time in Beijing is is 12 hours ahead.
8. Define on the set Z of integers the operation a∗b = a+b−2. Is Z with this operation a group?
9. Let G be a finite group, and consider the multiplication table for multiplication of G. Prove that
every element of G occurs precisely once in each row and once in each column.
10. Find a multiplication table on the set A = {a, b, c, d} where every element of A occurs precisely once in each row and once in each column but where A is not a group. You have to explain why your algebra A = {a, b, c, d} is not a group.

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