discrete mathematics

Sean E. Corpuz
Discrete Mathematics
January 13, 2004
Dr. Hairong Kuang
Homework #1
5. Several forms of negation are given for each of the following statements. Which are correct?
a. The answer is either 2 or 3.
1. Neither 2 nor 3 is the answer.
2. The answer is not 2 or not 3.
3. The answer is not 2 and it is not 3.
Answer: In this particular situation there are two forms which negate the statement
above. They are: 1 and 3.
b. Cucumbers are green and seedy.
1. Cucumbers are not green and not seedy.
2. Cucumbers are not green or not seedy.
3. Cucumbers are green and not seedy,
Answer: In this case, 1, 2, and 3 all negate the statement cucumbers are green AND
seedy.
c. 2 < 7 and 3 is odd.
1. 2 > 7 and 3 is even.
2. 2 ≥ 7 and 3 is even.
3. 2 ≥ 7 or 3 is odd.
4. 2 ≥ 7 or 3 is even.
Answer: Finally statements 1, 2, 3 and 4 all negate the primary statement above.
13. Using letters for component statements, translate the following compound
statements into symbolic notation.
a. If Anita wins the election, then tax rates will be reduced.
Answer:
A: Anita wins the election.
B: Tax rates will be reduced.
A → B

b. Tax rates will be reduced only if Anita wins the election and the economy
remains strong.
Answer:
A: Tax rates will be reduced.
B: Anita wins the election.
C: The economy remains strong.
(B ^ C) → A
c. Tax rates will be reduced, if the economy remains strong.
Answer:
A: Tax rates will be reduced.
B: The economy remains strong.
B → A
d. A strong economy will follow from Anita winning the election.
Answer:
A: A strong economy will follow.
B: Anita winning the election.
B → A
e. The economy will remain strong if and only if Anita wins the election or tax rates
are reduced.
Answer:
A: The economy will remain strong.
B: Anita wins the election.
C: Tax rates are reduced.
(B \/ C) ↔ A
14. Construct truth tables for the following well formed formulae. Note any tautologies or
contradictions.
c. A /\ (A’ \/ B’)
Answer:
A B A’ B’ (A’ \/ B’) A /\ (A’ \/ B’)
0 0 1 1 1 0

0 1 1 0 1 0
1 0 0 1 1 1
1 1 0 0 0 0
This well formed formula contains no tautology or contradiction whatsoever.
e. (A → B) → [(A \/ C) → (B \/ C)]
Answer:
A B C (A → B) (A \/ C) (B \/ C) (A → B) → [(A \/ C) → (B \/ C)]
0 0 0 1 0 0 1
0 0 1 1 1 1 1
0 1 0 1 0 1 1
0 1 1 1 1 1 1
1 0 0 0 1 0 1
1 0 1 0 1 1 1
1 1 0 1 1 1 1
1 1 1 1 1 1 1
Please take note that this particular well formed formula is a tautology.
g. A /\ B ↔ B’ \/ A’
Answer:
A B A’ B’ A /\ B B’ \/ A’ A /\ B ↔ B’ \/ A’
0 0 1 1 0 1 0
0 1 1 0 0 1 0
1 0 0 1 0 1 0
1 1 0 0 0 0 0
Please take note that this well formed formula is a contradiction with any value.
i. [(A \/ B) /\ C’] → A’ \/ C
Answer:
A B C A’ B’ C’ (A \/ B) A’ \/ C (A \/ B) /\ C’
0 0 0 1 1 1 0 1 0
0 0 1 1 1 0 0 1 0
0 1 0 1 0 1 0 1 0

0 1 1 1 0 0 0 1 0
1 0 0 0 1 1 0 0 0
1 0 1 0 1 0 0 0 0
1 1 0 0 0 1 1 0 1
1 1 1 0 0 0 1 1 0
[(A \/ B) /\ C’] → A’ \/ C
1
1
1
1
1
1
0
1
17.Verify each with a truth table:
d. A → A \/ B
Answer:
A B A \/ B A → A \/ B
0 0 0 1
0 1 1 1
1 0 1 1
1 1 1 1
Please note that this is a tautology.
f. (A /\ B)’ ↔ A’ \/ B’
A B A’ B’ (A /\ B)’ A’ \/ B’ (A /\ B)’ ↔ A’ \/ B’
0 0 1 1 1 1 1
0 1 1 0 1 1 1
1 0 0 1 1 1 1
1 1 0 0 0 0 1
Note that this well formed formula is a tautology.

Of all the statements which were given to us, only four are propositions. They are the following:
c. There are no black flies in Maine.
In this case, the truth value of this statement lies within the fact that there are no black
flies in Maine.
d. 4 + x = 5
The truth value for this particular statement is x = 1.
e. x + 1 = 5 if x =1
This statement with the given values is false; however, it doesn’t negate the fact that this
is a preposition.
f. x + y = y + z if x = z
In order for this statement to be considered truth, you need the value of x to equal z. thus
demonstrating the commutative property.