# Sample Exam I by alwaysnforever

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```									                   Math 302—Sample EXAM I

You must show your work to get credit.

1. 12 pts Construct a truth table and determine whether or not each of
the following is a tautology, contigency or contradiction.

(a) (p → q) ∨ p

(b) p → (q ∧ ¬r)

2. 12 pts Given the following premises list the conclusions that can be
drawn and explain the rules of inference used to obtain the conclusionts:
“Every computer science major has a PC.” “Ralph does not have a
PC.” “Ann has a PC.”

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3. 12 pts Prove that A ∩ (A ∪ B) = A.

4. 12 pts Draw a Venn diagram to describe the set A ∪ (B ∩ C).

5. 14 pts Prove that if n is a positive integer n is even if and only if 7n +
4 is even.

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6. 14 pts

(a) Using at least two quantiﬁed variables express the following state-
ment in symbolic form: “All students in this class have learned at
least one programming language.”

(b) Write symbolically a simpliﬁed negation of the above expressiona
and then translate this negated expression into English.

7. 15 pts Determine which of the following functions are injective and
which are surjective. Also determine the range of each function.

(a) f : R → R given by f (x) = x2 + 2.

(b) f : R → R given by f (x) = −3x + 4.

(c) f {0, 1, 2} → {0, 1, 2, 3, 4} given by f (0) = 3, f (1) = 4, and f (2) =
0.

8. 9 pts Deﬁne Proof by Contraposition.

9. (9 pts) Show that 3n2 + 5n + 42 ∈ Θ(n2 ).

10. (9 pts) Give an example of a function f : Z+ → R where f ∈ Ω(n),
but f ∈ Θ(n). Explain why.

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