# De Morgan’s laws

Use De Morgan’s laws to find the negation of the following statement.

Kwame will take a job in industry or go to graduate school.

 Kwame will not take a job in industry and will not go to graduate school. Kwame will take a job in industry and will not go to graduate school. Kwame will not take a job in industry and will go to graduate school. Kwame will take a job in industry and will go to graduate school. Kwame will not take a job in industry or will not go to graduate school.

Complete the truth table for each of these compound propositions.

a) p→(¬qr)

 p q r ¬q ¬q∧r p→(¬q∧r) T T T T T F T F T T F F F T T F T F F F T F F F

b) ¬p→(qr)

 p q r ¬p q→r ¬p→(q→r) T T T T T F T F T T F F F T T F T F F F T F F F

c) (pq)∧(¬pr)

 p q r ¬p p→q ¬p→r (p→q)∧(¬p→r) T T T T T F T F T T F F F T T F T F F F T F F F

d) (pq)∧(¬qr)

 p q r ¬q p↔q ¬q↔r (p↔q)∧(¬q↔r) T T T T T F T F T T F F F T T F T F F F T F F F

e) p↔¬q)↔(qr)

 p q r ¬p ¬q ¬p↔¬q q↔r (¬p↔¬q)↔(q↔r) T T T T T F T F T T F F F T T F T F F F T F F F

Show that each of these conditional statements is a tautology by completing the truth tables.

a) (pq)→p

 p q p∧q (p∧q)→p T T T F F T F F

b) q→(pq)

 p q p∨q q→(p∨q) T T T F F T F F

c) ¬p→(pq)

 p q ¬p p→q ¬p→(p→q) T T T F F T F F

d) (pq)→(pq)

 p q p∧q p→q (p∧q)→(p→q) T T T F F T F F

e) ¬(pq)→p

 p q p→q ¬(p→q) ¬(p→q)→p T T T F F T F F

f) ¬(pq)→¬q

 p q p→q ¬(p→q) ¬q ¬(p→q)→¬q T T T F F T F F
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