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MATH 302.903
Practice Problems for Examination 1
Spring 2006
1. For each of the following sentences, indicate whether it is a proposition, and in the case
that it is a proposition also indicate its truth value.
(a) 7 divides 13n ?1.
(b) A function f : R ? R is differentiable if and only if f(0) is a rational number.
(c) For every even integer n, 12n3 ?2n2 + 18 is even.
(d) ? ? ?.
(e) ? ? A.
(f) f(x) is of order ex.
(g) floorleftpifloorright = 3.
2. Let n and m be positive integers. Consider the following proposition P: "If n and
m are both divisible by 3 then n(m + 1) is even." Write the converse, inverse, and
contrapositive of P.
3. Incorporate each of the following sentences into a false proposition.
(a) ? ? A.
(b) f is bounded.
(c) |A| = 8.
(d) Every differentiable function f : R ? R is continuous.
(e) A ? (B ?C).
4. Determine, with proof, the least integer n such that x4 lnx?3x2 + 4 is O(xn).
1
5. Determine which of the following pairs of compound propositions are logically equiva-
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