Show $[(P\Rightarrow Q)\land(Q\Rightarrow R)]\Rightarrow(P\Rightarrow R)$ is a tautology using a truth table.

Show $$[(P\Rightarrow Q)\land(Q\Rightarrow R)]\Rightarrow(P\Rightarrow R)$$ is a tautology using a truth table.

I have constructed the following truth table, however, I must have made an error because there are two rows that do not actually match between the hypothesis and conclusion. Could someone point out where the problem is?

$$P$$ $$Q$$ $$R$$ $$P\Rightarrow Q$$ $$Q\Rightarrow R$$ $$(P\Rightarrow Q)\land(Q\Rightarrow R)$$ $$(P\Rightarrow R)$$ $$[(P\Rightarrow Q)\land(Q\Rightarrow R)]\Rightarrow (P\Rightarrow R)$$
T T T T T T T T
T T F T F F F T
T F F F T F F T
T F T F T F T T
F T T T T T T T
F F T T T T T T
F F F T T T T T
F T F T F F T T

The problem is with my fourth and last row. I where the entries in the last two columns do not align.

• $F \Rightarrow T$ is valued as true. So those rows are not misaligned. Feb 8, 2022 at 1:29

You were not asked to show that $$(P \implies Q) \land (Q \implies R)$$ is equivalent to $$P \implies R$$. Rather, you were asked to show that $$((P \implies Q) \land (Q \implies R)) \implies (P \implies R)$$ is a tautology. So, you need to make a column for the proposition $$((P \implies Q) \land (Q \implies R)) \implies (P \implies R)$$ and verify that each entry in the column is T.