MAT 185 – 001 Spring 2016 Test one Show all work

Theorem 2.1.1 Logical Equivalences

p, q, and r are statement variables; t is a tautology; and c is a contradiction.

1. Commutative laws: pqqp∧≡∧ pqqp∨≡∨

2. Associative laws: ( pq∧∧≡∧∧) r p( qr) ( pq∨∨≡∨∨) r p( qr)

3. Distributive laws: p∧∨≡( qr) ( pq ∧∨) ( pr ∧) p∨∧≡( qr) ( pq ∨∧∨) ( pr)

4. Identity laws: pp∧≡t pp∨≡c

5. Negation laws: pp∨≡ t pp∧≡ c

6. Double negative law: ( pp) ≡

7. Idempotent laws: ppp∧≡ ppp∨≡

8. Universal bound laws: p ∨≡tt p ∧≡cc

9. De Morgan’s laws: ( pq∧≡)  p ∨ q ( pq∨≡)  p ∧ q

10. Absorption laws: p∨∧≡( pq) p p∧∨≡( pq) p

11. Negations of t and c:  tc≡  ct≡

Valid Argument Forms

Modus Ponens pq→ p ∴q

Modus Tollens pq→  q ∴ p

Generalization

a. pqb.

∴∨pq ∴∨ pq

Specialization a.pq∧∧ b. pq

∴∴pq

Conjunction p q ∴∧pq

Elimination a.pq∨∨ b. pq qp ∴∴pq

Transitivity pq→ qr→ ∴→pr

Proof by Division into Cases pq∨ pr→

qr→ ∴r

Contradiction rule

 pc→

∴ p

1. Write the negation for the following statement:

Part A has failed or part B has failed.

2. Consider the following statement:

If m is even then m2 is even.

2a. Write the negation of the statement.

2b. Write the contrapositive of the statement.

2c. Write the converse of the statement.

2d. Write the inverse of the statement.

3. Construct a truth table for the following: p∧ q ∧∨( pr)

Label all columns.

pqr FFF FFT FTF FTT TFF TFT TTF TTT

3. Use a truth table to determine whether the following argument is valid or not. Indicate which columns represent premises and which column represents the conclusion. Indicate the critical rows. Explain how you determined that the argument is valid or not.

pq∧→ r pq→

p ∴r

pqr FFF FFT FTF FTT TFF TFT TTF TTT

4. Use theorem 2.1.1 to prove the following logical equivalence. Supply a reason for each step.

( pq→) ≡( pq →) ∧∨( r r) .

5. Use truth tables to prove the logical equivalence:

( pq∧→∨≡) ( pq) t

where t is a tautology.

Explain how the table is used to prove the equivalence. Label all columns.

pq FF

FT TF TT

6. Without using a truth table (i.e., using the valid argument forms), show that the following is a valid argument. Justify each step as one of the valid argument forms.

pq→ qr→ r  sp∨ st→ ∴t

t is a statement – not a tautology.

7a. Use symbols to write the logical form of the following argument.

All cyclists wear helmets. Gomer doesn’t wear a helmet. Therefore, Gomer isn’t a cyclist.

7b. Determine whether this is a valid argument or not. Explain your answer.