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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.

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

1 Answer 1

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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.

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