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CMSC 203 - Discrete Structures - Fall 1993 - Examination 1
1. (16 pts.) Construct the truth table for the following statement: [~r → (p ∨ ~q)] ∨ s
2. (14 pts.) Prove:
a. Using truth tables that: b. Without using truth tables that:
~(p → q) ≡ p ∧ ~q ~[~(r ∧ s) ∨ (p ∧ ~q)] ≡ (r ∧ s) → (p ∧ ~q)
(hint: use part a)
3. (20 pts.)
a. Find the truth table of the b. Find the Boolean polynomial whose truth
Boolean function f(x,y) = x + xy table is:
x y f(x,y) x y z f(x,y,z)
1 1 1 1 1 0
1 0 1 1 0 0
0 1 1 0 1 1
0 0 1 0 0 0
0 1 1 1
0 1 0 0
0 0 1 0
0 0 0 0
c. Without using truth tables, prove xy + x’y + x’y’ = x’ + y .
4. (10 pts.) Find the circuit diagram for a system of three light switches which turn a light ON
only when all three switches are either all ON or all OFF.
5. (20 pts.) a. Rewrite the statement: All integers divisible by 2 are even into the form:
For all ___, if ___, then ___.
b. Rewrite the statement: Some primes are even into the form:
There exists ___, such that ___.
c. Express the statement: "∀x ∈ R, x2 ≥ 0" in simple English.
d. Write the negation of the statement: All Computer Science students hate math.
e. Write the negation of the statement: ∀x ∈ R, x > 7 implies x2 > 49.
6. (5 pts.) Disprove by counterexample the statement:
For all prime numbers, k, (-1)k < 0.
7. (15 pts.) Prove the statement: The sum of any even and any odd integer is odd.
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