discrete mathematics

Answer: In this particular situation there are two forms which negate the statement above.. Cucumbers are green and not seedy, Answer: In this case, 1, 2, and 3 all negate the statement

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:

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:

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’

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