MATHEMATICS

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 ¬qr p→(¬qr)
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 qr ¬p→(qr)
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 pq ¬pr (pq)∧(¬pr)
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 pq ¬qr (pq)∧(¬qr)
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 qr p↔¬q)↔(qr)
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 pq (pq)→p
T T
T F
F T
F F

 

b) q→(pq)

p q pq q→(pq)
T T
T F
F T
F F

 

c) ¬p→(pq)

p q ¬p pq ¬p→(pq)
T T
T F
F T
F F

 

d) (pq)→(pq)

p q pq pq (pq)→(pq)
T T
T F
F T
F F

 

e) ¬(pq)→p

p q pq ¬(pq) ¬(pq)→p
T T
T F
F T
F F

 

f) ¬(pq)→¬q

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

 

 

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