Question

.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)

Answer 1

See explanation below.

Step-by-step 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

`(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 since for all the possibilites we got true the final result.

Part b

`p \Rightarrow (r \Rightarrow p)` let's call this condition (1)

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