Question

Use the laws of propositional logic to prove that each statement is a tautology. (p n q) rightarrow (p V r) p rightarrow (r rightarrow p) (8 points each for a total of 16, zyBook section 1.5, exercise 1.5.3(a, b))

Answer 1

Answer:

See explanation below.

Explanation:

If the statement is a tautology is true for all the possible combinations

Part a

$(p \land q) \Rightarrow (p \lor r)$ lets call this condition (1)

$(p \land q)$ condition (2) and $(p \lor r)$ condition (3)

We can create a table like this one:

p       q     r      (2)       (3)     (1)  

T       T     T      T        T       T

T       T     F      T        T       T        

T       F     T      F        T       T

T       F     F      F        T       T

F       T     T      F        T       T

F       T     F      F        F       T

F       F     T      F        T       T

F       F     F      F        F       T

So as we can see we have a tautology.

Part b

$p \Rightarrow (r \Rightarrow p)$ lt's call this condition 1

And $(r \Rightarrow p)$ condition 2

We can create the following table:

p     r       (2)     (1)

T     T       T       T

T     F       T       T

F     T       F       T

F     F       T       T

So is also a tautology.