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.
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Here are the solutions for questions 3, 4, and 5. 3) Construct a truth table for the following compound propositions: a) (P Q) (P Q) | P | Q | P Q | P Q | (P Q) (P Q) | | :--: | :--: | :--------: | :--------: | :---------------------------------: | | T | T | T | T | T | | T | F | T | F | F | | F | T | T | F | F | | F | F | F | F | T | b) (P Q) ( Q P) | P | Q | P | Q | P Q | Q P | (P Q) ( Q 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 Q P Q This argument is invalid. | P | Q | P Q | P | 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 Q and P) are true, but the conclusion ( 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 Q P Q This argument is valid. | P | Q | P Q | 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 Q and 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) (P Q) ( P Q) Step 1: Apply De Morgan's Law to the first term. (P Q) ( P ( Q)) ( P Q) Step 2: Substitute back into the original expression. ( P Q) ( P Q) Step 3: Apply the Distributive Law. P (Q Q) Step 4: Apply the Complement Law (Q Q T). P T Step 5: Apply the Identity Law ( P T P). P b) (P Q) (P Q) Step 1: Replace implications with disjunctions using A B A B. ( P Q) ( P Q) Step 2: Apply the Distributive Law. P (Q Q) Step 3: Apply the Complement Law (Q Q F). P F Step 4: Apply the Identity Law ( P F P). P That's 2 down. 3 left today — send the next one.