1 and 2 I mean, what about 3 4 5

Here are the solutions for questions 3, 4, and 5.

3) Construct a truth table for the following compound propositions:

• a) $(P \lor Q) \Rightarrow (P \land Q)$

$P$	$Q$	$P \lor Q$	$P \land Q$	$(P \lor Q) \Rightarrow (P \land Q)$
T	T	T	T	T
T	F	T	F	F
F	T	T	F	F
F	F	F	F	T

• b) $(P \Rightarrow Q) \Leftrightarrow (\sim Q \Rightarrow \sim P)$

$P$	$Q$	$\sim P$	$\sim Q$	$P \Rightarrow Q$	$\sim Q \Rightarrow \sim P$	$(P \Rightarrow Q) \Leftrightarrow (\sim Q \Rightarrow \sim P)$
T	T	F	F	T	T	T
T	F	F	T	F	F	T
F	T	T	F	T	T	T
F	F	T	T	T	T	T

4) Using truth tables, verify whether the following arguments are valid or invalid:

• a) $P \Rightarrow Q$ $\sim P$ $\therefore \sim Q$

This argument is invalid.

$P$	$Q$	$P \Rightarrow Q$	$\sim P$	$\sim Q$
T	T	T	F	F
T	F	F	F	T
F	T	T	T	F
F	F	T	T	T

• Explanation: An argument is valid if, whenever all premises are true, the conclusion is also true. In the third row, both premises ($P \Rightarrow Q$ and $\sim P$) are true, but the conclusion ($\sim Q$) is false. This is a counterexample, showing the argument is invalid. This is an example of the Fallacy of Denying the Antecedent.

• b) $P \lor Q$ $\sim P$ $\therefore Q$

This argument is valid.

$P$	$Q$	$P \lor Q$	$\sim P$	$Q$
T	T	T	F	T
T	F	T	F	F
F	T	T	T	T
F	F	F	T	F

• Explanation: We look for rows where all premises ($P \lor Q$ and $\sim P$) are true. This occurs only in the third row. In this row, the conclusion ($Q$) is also true. Therefore, the argument is valid. This is a valid form of argument known as Disjunctive Syllogism.

5) Using the laws of logic, simplify the following compound propositions:

• a) $\sim (P \lor \sim Q) \lor (\sim P \land \sim Q)$