PARTA
Each question is worth 2 points.
DO NOT INCLUDE PROOFS OR COMMENTS FOR YOUR WORK IN THIS SECTION.
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1. “Lansing is the capital of Michigan” and “It never rains in
Phoenix.” are statements that are True.
True or False
2. The compound statement ” p ^ (p v q) -> q ” is False only when p
is False.
True or False
3. The disjunction of p and q is the same as (p v q).
True or False
4. An OR function is the inverse of a NOR function.
True or False
5. The truth table of the biconditional statement, p <=> q , is the
inverse of p XOR q
True or False.
6. A proof that tries all possible cases to determine the result is
termed an exhaustive proof.
True or False
7. The converse of the contrapositive of a proposition is the same as
the original statement.
True or False
8. p NOR p = p AND p only when p = F.
True or False
9. The proposition (p OR FALSE) is True only when p is TRUE.
True or False
10. The contrapositive of a proposition is the inverse of the
original statement.
True or False
11. The relation
NOT (q OR p) AND (q NOR NOT p)
is True only when q is FALSE.
True or False
12. A statement that is neither a tautology nor a contingency is
termed a contrdiction.
True or False
13. Lemmas and axioms are both used to construct a proof.
True or False
14. The sum of the first 42 even integers is 1806.
True or False
PART B
Each question is worth 6 points. Divided questions are worth 3 points for each section. Show all work (within reason) in intermediate steps. Partial credit will be given, as appropriate.
Note: ANSWERS WITHOUT INTERMEDIATE WORK WILL BE GRADED AS ZERO!
Clearly identify the answer.
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1. Generate the Truth Table for
(p ^ s) ^ (NOT p ^ s)
2. How many rows will a Truth Table require if there are five
conditions and three variables? Defend your answer.
3. Construct a Truth Table for the statement NOT p -> p
4. Construct a Truth Table for the statement “p XOR (p v q)”
5. Given the bit strings 1111 1000 and 1001 1011, define the
a. bitwise AND
b. bitwise OR
6. Let:
A = {1,2,3,4,5,6,7,10,11)
B = {1,4,5,6,8,9,11}
C = {1,3,4,5,7,9,10}
Find the bitwise:
a. A XOR B
b. C DIFF A
7. Using the provided statement, what are the associated logic and prose statements for the:
a) converse; and,
b) contrapositive of the statement:
“If it is not Monday night, I will go to the movies”.
8. Using a Truth Table or equivalent, determine if:
(q AND NOT p) and (NOT(NOT q OR p)) are equivalent.
9. Let:
A = {a,b,f,g,h,i,l},
B = {b,c,d,f,g,j,m}, and
C = {a,c,d,f,g,i,j,l,m}.
Define each of the following bitwise operations:
a) (A NOR B) OR C
b) (A – B) – C
10. Determine whether p v NOT( p ^ q) -> q is True only when q is
False.
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OPTIONAL EXTRA CREDIT QUESTION (2 Points or no points). DO ONE.
A. Provide the Basis Step (ONLY) for:
What is the sum of the first n odd positive integers?
Defend your answer with six examples.
B. Is the following statement a Tautology? If not, what?
[(p 6 q) ^ (p -> r) v (q -> r) } -> r
Defend your answer.
END.
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