DM A5 03052020 095913 pm PDF

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Bahria University, Islamabad Campus Department of Computer Sciences Discrete Math Assignment 5 Due Date:3/5/20 1. Determine whether the statement forms are logically equivalent. In each case, construct a truth table. p ∨ (p ∧ q) and p Answer:

(p ∧ q) ∧ r and p ∧ (q ∧ r) Answer:

2. Use truth tables to establish which of the statement forms are tautologies and which are contradictions. a. (p ∧ q) ∨ (∼p ∨ (p ∧ ∼q))

b. (p ∧ ∼q) ∧ (∼p ∨ q)

3. Assume x is a particular real number and use De Morgan’s laws to write negations for the statements. a. −2 < x < 7

b. x < 2 or x > 5

4.Using DeMorgan’s rule negate the following propositions. a. Sam is an orange belt and Kate is a red belt. Answer: Sam is not an orange belt or Kate is not a red belt. c. The connector is loose or the machine is unplugged Answer: The Connector is not loose and the machine is not unplugged.

5.Simplify the following using rules of logic. 1) ~(~p ∧ q) ∧ ( p v q)

3) (p ∨ ∼q) ∧ (∼p ∨ ∼q)

5) p ∧ (∼q ∨ p)

2) (p ∧ ∼q) ∨ (p ∧ q)

4) (p ∧ ∼q) ∨ p

6)∼((∼p ∧ q) ∨ (∼p ∧ ∼q)) ∨ (p ∧ q)

7) (p ∧ (∼(∼p ∨ q))) ∨ (p ∧ q)

8) p →(~p v q)

9) (p v q) →(~p ∧ q)

6. Use a Truth table to check if below mentioned propositions form a tautology a)

b)

7. Write the negation of the following: a. If the connector is l oose the machine will not work. Answer: The Connector is loose and the machine will work. b. If Sara is orange belt then Sara’s sister is red belt. Answer:

Sara is Orange belt and Sara’s Sister is no red belt. 8. Write the inverse, converse and contraposit ive of the following statement. a. If Ahmed lives in Lahore then Ahmed lives in Pakistan. INVERSE: If Ahmed will not live in Lahore then Ahmed will not live in Pakistan. CONVERSE: If Ahmed Lives in Pakistan Ahmed Lives in Lahore. CONTRA-POSITIVE: If Ahmed does not live in Pakistan Ahmed does not live in Lahore

9. Implication is logically equivalent to ___________________ (inverse, converse, contrapositive). Answer: Contra-Positive...