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```MATH 260 2001/2002 Midterm Exam 1
November 9, 2001
Dr. Bing Zhou
Name
Instructions. All answers should be clear and complete. Show all your works. Partial credit will be given
1. (5 points)
(a) Determine whether the following statement is true or false (explain why briefly). The universe of
discourse is R, the set of all real numbers.
1. ∃x∀y (y > x) .
´
³
2
2. ∃x∀y (x + y) = x2 + y2 .
(b) Find the converse and contrapositive of the statement “x is an even number is the necessary
condition for x to be a multiple of 4”.
2. (5 points) Let A be a set of eight elements and B a set of fifteen elements. How many functions from
A to B are not one-to-one?
3. (5 points) Let x, y, and z be three integers such that xy = z. Prove that if x and z are odd numbers,
y must be an odd number.
2
4. (5 points) Use mathematical induction to prove that (n + 1) ≤ 2n2 for all n ≥ 3.
2
```
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