Read about Cantor's diagonal argument. He used it to show there were different sizes of infinite numbers. Explain how it is done and what number sets it and which were different.

Give the statement of the Fundamental Theorem of Arithmetic.

Another Field We'll let Q(√2) represent the set of numbers that can be written in the form a+b√2, where 'a' and 'b' and rational numbers (i.e., 'a' and 'b' are from 13 2 the set Q). For example, +¹3√2, 7.2-6√2, and 15 (= 15 + 0√2) are three numbers that would be in the set Q(√2), but √√3+ 4√2 or 3 + √2 would not be in the set Q(√2) (neither √√3 nor are rational numbers). A number such 3+5√2 would be in the set Q(√2) since 3+√2 can be re-written in the form 7 a+b√2 using the following steps: 7 7 3-5√2 21-35√2 21-35√2 21 - 35√2 21 35 == + √2 3+5√2 3+5√2 3-5√2 9 - 25(2) -41 41 41 = * Your task: Show that the system (Q(√2), +, *) is a field, where + is the ordinary addition operation and * is the ordinary multiplication operation. So, you need to show that: a. (Q(√2), +, *) is a commutative ring, and b. (Q(√2) - {0}, *) is a group.

True or False: For all integers a, b, and c, if a divides b and b divides c, then a divides c. True O False