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題組內容

3. Let be sets from a uni verse .

(a) Wiite a quantified statement to express the proper subset relation .

其他申論題

8.Let with . Prove that if and , then .

(a) there must be seven men and seven women?

(b) there must be at least eight men?

2. Verify that , for primitive statements , and .

(b) Negate the result in part (a) to determine when .

4. (a) Consider an chessboard. It contains eighty-one squares and one square. How many squares?

(b) Now consider an chessboard for some fixed . ForI , how many squares are contained in this chessboard?

5. Let be a set of five positive integers the maximum of which is at most 9. Prove that the sums of the elements in all the nonempty subsets of S cannot all be distinct.

6. (a) Fermat's Theorem. If is a prime, prove that for each .

(b) Buler's Theorem. For each n Ezt,n ＞ 1, and each a EZ, prove that if gcd(a,n) = 1, then (a中(n) = 1(mod n).

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