We are given a truth table with columns for the propositions P, Q, and R. We need to complete the table by evaluating the expressions $P \rightarrow Q$, $Q \rightarrow R$, $P \land Q$, and $(P \rightarrow Q) \rightarrow (Q \rightarrow R)$ for all possible truth values of P, Q, and R.

1. Problem Description

We are given a truth table with columns for the propositions P, Q, and R. We need to complete the table by evaluating the expressions $P \rightarrow Q$, $Q \rightarrow R$, $P \land Q$, and $(P \rightarrow Q) \rightarrow (Q \rightarrow R)$ for all possible truth values of P, Q, and R.

2. Solution Steps

We will use the following truth tables for implication ($\rightarrow$) and conjunction ($\land$):

$P \rightarrow Q$ is false only when P is true and Q is false. Otherwise, it is true.

$P \land Q$ is true only when both P and Q are true. Otherwise, it is false.

| P | Q | $P \rightarrow Q$ | P | Q | $P \land Q$ |

|---|---|--------------------|---|---|---------------|

| T | T | T | T | T | T |

| T | F | F | T | F | F |

| F | T | T | F | T | F |

| F | F | T | F | F | F |

Now, we can complete the truth table step by step.

Row 1: P=T, Q=T, R=T

$P \rightarrow Q = T \rightarrow T = T$

$Q \rightarrow R = T \rightarrow T = T$

$P \land Q = T \land T = T$

$(P \rightarrow Q) \rightarrow (Q \rightarrow R) = T \rightarrow T = T$

Row 2: P=T, Q=T, R=F

$P \rightarrow Q = T \rightarrow T = T$

$Q \rightarrow R = T \rightarrow F = F$

$P \land Q = T \land T = T$

$(P \rightarrow Q) \rightarrow (Q \rightarrow R) = T \rightarrow F = F$

Row 3: P=T, Q=F, R=T

$P \rightarrow Q = T \rightarrow F = F$

$Q \rightarrow R = F \rightarrow T = T$

$P \land Q = T \land F = F$

$(P \rightarrow Q) \rightarrow (Q \rightarrow R) = F \rightarrow T = T$

Row 4: P=T, Q=F, R=F

$P \rightarrow Q = T \rightarrow F = F$

$Q \rightarrow R = F \rightarrow F = T$

$P \land Q = T \land F = F$

$(P \rightarrow Q) \rightarrow (Q \rightarrow R) = F \rightarrow T = T$

Row 5: P=F, Q=T, R=T

$P \rightarrow Q = F \rightarrow T = T$

$Q \rightarrow R = T \rightarrow T = T$

$P \land Q = F \land T = F$

$(P \rightarrow Q) \rightarrow (Q \rightarrow R) = T \rightarrow T = T$

Row 6: P=F, Q=T, R=F

$P \rightarrow Q = F \rightarrow T = T$

$Q \rightarrow R = T \rightarrow F = F$

$P \land Q = F \land T = F$

$(P \rightarrow Q) \rightarrow (Q \rightarrow R) = T \rightarrow F = F$

Row 7: P=F, Q=F, R=T

$P \rightarrow Q = F \rightarrow F = T$

$Q \rightarrow R = F \rightarrow T = T$

$P \land Q = F \land F = F$

$(P \rightarrow Q) \rightarrow (Q \rightarrow R) = T \rightarrow T = T$

Row 8: P=F, Q=F, R=F

$P \rightarrow Q = F \rightarrow F = T$

$Q \rightarrow R = F \rightarrow F = T$

$P \land Q = F \land F = F$

$(P \rightarrow Q) \rightarrow (Q \rightarrow R) = T \rightarrow T = T$

3. Final Answer

Here is the completed truth table:

| P | Q | R | $P \rightarrow Q$ | $Q \rightarrow R$ | $P \land Q$ | $(P \rightarrow Q) \rightarrow (Q \rightarrow R)$ |

|---|---|---|--------------------|--------------------|---------------|---------------------------------------------------|

| T | T | T | T | T | T | T |

| T | T | F | T | F | T | F |

| T | F | T | F | T | F | T |

| T | F | F | F | T | F | T |

| F | T | T | T | T | F | T |

| F | T | F | T | F | F | F |

| F | F | T | T | T | F | T |

| F | F | F | T | T | F | T |