Using the laws of logic to prove tautologies. use the laws of propositional logic to prove that each statement is a tautology. (a) (p ∧ q) → (p ∨ r) (b) p → (r → p)

See explanation below.

Explanation:

If the statement is a tautology is true for all the possible combinations and we can check this with the table of truth for the statements

Part a

lets call this condition (1)

condition (2) and  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 since for all the possibilites we got true the final result.

Part b

let's call this condition (1)

And let 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 since the statement is true for all the possibilities or combinations.

df =

both have same length