Problem- 26

Problem- 26

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

We conclude that each of these conditional statements is a tautology because the entries in the last column contain ………..?

a) (pq)→p

b) p→(pq)

c) ¬p→(pq)

d) (pq)→(pq

e) ¬(pq)→p

f) ¬(pq)→¬q

 

Solution
a) (pq)→p

 

p

q

pq

(pq)→p

T

T

  T

  T

T

F

  F

  T

F

T

  F

  T

F

F

  F

  T

 

b) p→(pq)

 

p

q

pq

p→(pq)

T

T

  T

  T

T

F

  T

  T

F

T

  T

  T

F

F

  F

  T

 

c) ¬p→(pq)

 

p

q

¬p

pq

¬p→(pq)

T

T

  F

  T

  T

T

F

  F

  F

  T

F

T

  T

  T

  T

F

F

  T

  T

  T

 

d) (pq)→(pq)

 

p

q

pq

pq

(pq)→(pq)

T

T

  T

  T

  T

T

F

  F

  F

  T

F

T

  F

  T

  T

F

F

  F

  T

  T

 

e) ¬(pq)→p

 

p

q

pq

¬(pq)

¬(pq)→p

T

T

  T

  F

  T

T

F

  F

  T

  T

F

T

  T

  F

  T

F

F

  T

  F

  T

 

f) ¬(pq)→¬q

 

p

q

pq

¬(pq)

¬q

¬(pq)→¬q

T

T

  T

  F

  F

  T

T

F

  F

  T

  T

  T

F

T

  T

  F

  F

  T

F

F

  T

  F

  T

  T

We conclude that each of these conditional statements is a tautology because the entries in the last column contain all Ts.

 

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