1 and 2 I mean, what about 3 4 5
This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
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) $\sim (P \lor \sim Q) \lor (\sim P \land \sim Q)$
Step 1: Apply De Morgan's Law to the first term.
$$ \sim (P \lor \sim Q) \equiv (\sim P \land \sim (\sim Q)) \equiv (\sim P \land Q) $$
Step 2: Substitute back into the original expression.
$$ (\sim P \land Q) \lor (\sim P \land \sim Q) $$
Step 3: Apply the Distributive Law.
$$ \sim P \land (Q \lor \sim Q) $$
Step 4: Apply the Complement Law ($Q \lor \sim Q \equiv T$).
$$ \sim P \land T $$
Step 5: Apply the Identity Law ($\sim P \land T \equiv \sim P$).
$$ \boxed{\sim P} $$
b) $(P \Rightarrow Q) \land (P \Rightarrow \sim Q)$
Step 1: Replace implications with disjunctions using $A \Rightarrow B \equiv \sim A \lor B$.
$$ (\sim P \lor Q) \land (\sim P \lor \sim Q) $$
Step 2: Apply the Distributive Law.
$$ \sim P \lor (Q \land \sim Q) $$
Step 3: Apply the Complement Law ($Q \land \sim Q \equiv F$).
$$ \sim P \lor F $$
Step 4: Apply the Identity Law ($\sim P \lor F \equiv \sim P$).
$$ \boxed{\sim P} $$
That's 2 down. 3 left today — send the next one.