Interactive Predicate Logic

Solution Explanation:

Step 1: Calculate ¬P (negation of P)

• When P = T, ¬P = F
• When P = F, ¬P = T

Step 2: Calculate ¬P ∨ Q (¬P OR Q)

• Row 1: F ∨ T = T
• Row 2: F ∨ F = F
• Row 3: T ∨ T = T
• Row 4: T ∨ F = T

Solution Explanation:

The exclusive OR (XOR) operation is true when exactly one of the propositions is true, but not both.

Looking at the truth table:

• T ⊕ T = F (both true → false)
• T ⊕ F = T (exactly one true → true)
• F ⊕ T = T (exactly one true → true)
• F ⊕ F = F (both false → false)

Solution Explanation:

Let's evaluate step by step with P = F, Q = T, R = T:

1. P ∧ Q = F ∧ T = F (AND requires both to be true)
2. ¬P = ¬F = T (negation flips the value)
3. ¬P ∧ R = T ∧ T = T (both are true)
4. (P ∧ Q) ∨ (¬P ∧ R) = F ∨ T = T (OR is true if at least one operand is true)

The final result is T (True).

Solution Explanation:

De Morgan's Law states that the negation of a conjunction equals the disjunction of the negations.

Looking at each row:

• Row 1: ¬(T ∧ T) = ¬T = F, and (¬T ∨ ¬T) = (F ∨ F) = F ✓
• Row 2: ¬(T ∧ F) = ¬F = T, and (¬T ∨ ¬F) = (F ∨ T) = T ✓
• Row 3: ¬(F ∧ T) = ¬F = T, and (¬F ∨ ¬T) = (T ∨ F) = T ✓
• Row 4: ¬(F ∧ F) = ¬F = T, and (¬F ∨ ¬F) = (T ∨ T) = T ✓

Both expressions have identical truth values, proving they are equivalent!

Solution Explanation:

An implication P → Q is false only when the antecedent (P) is true and the consequent (Q) is false.

Truth table for P → Q:

• T → T = T (true premise, true conclusion)
• T → F = F (true premise, false conclusion - this is the only false case)
• F → T = T (false premise - implication is vacuously true)
• F → F = T (false premise - implication is vacuously true)

Think of it as: "If I study hard, then I will pass the exam." This statement is only false if I study hard but don't pass.

Solution Explanation:

With P = F, Q = F, R = T:

1. P ∨ Q = F ∨ F = F (OR needs at least one true value)
2. Q ∨ R = F ∨ T = T (R is true, so the OR is true)
3. (P ∨ Q) ∧ (Q ∨ R) = F ∧ T = F (AND needs both operands to be true)

The final result is F (False).

Solution Explanation:

To find ¬(P → Q), we need to understand when P → Q is false.

P → Q is false only when P is true and Q is false (T → F = F).

Therefore, ¬(P → Q) is true only when P is true and Q is false.

This matches the expression P ∧ ¬Q:

• When P = T and Q = F: P ∧ ¬Q = T ∧ T = T ✓
• For all other cases: P ∧ ¬Q = F ✓

So ¬(P → Q) ≡ P ∧ ¬Q

Solution Explanation:

Exportation Law: (P ∧ Q) → R ≡ P → (Q → R)

This law shows that "if both P and Q, then R" is equivalent to "if P, then (if Q, then R)".

Since both expressions are equivalent, the biconditional (↔) between them is always true!

The final result column should be all T's, demonstrating that this is a tautology - a statement that is always true regardless of the truth values of its components.

This is a fundamental law in logic that allows us to transform conditional statements.

Solution Explanation:

Distributive Law: P ∧ (Q ∨ R) ≡ (P ∧ Q) ∨ (P ∧ R)

This law shows that conjunction distributes over disjunction, similar to how multiplication distributes over addition in arithmetic.

Since both expressions are logically equivalent, they always have the same truth value, making the equivalence always true (tautology).

This is fundamental in simplifying complex logical expressions and circuit design.

Solution Explanation:

Contrapositive Law: (P → Q) ≡ (¬Q → ¬P)

The contrapositive of an implication is logically equivalent to the original statement.

Examples:

• Original: "If it rains, then the ground gets wet"
• Contrapositive: "If the ground is not wet, then it did not rain"

This is extremely useful in mathematical proofs - you can prove P → Q by proving ¬Q → ¬P instead!

Solution Explanation:

Absorption Laws:

• First Law: P ∨ (P ∧ Q) ≡ P
• Second Law: P ∧ (P ∨ Q) ≡ P

These laws show that certain complex expressions can be "absorbed" and simplified:

• When P is true, P ∨ (anything) = P, and P ∧ (anything involving P) = P
• When P is false, P ∨ (P ∧ Q) = F ∨ F = F = P, and P ∧ (P ∨ Q) = F ∧ (anything) = F = P

These are crucial for simplifying logical circuits and expressions in computer science and digital design!

Both equivalences are tautologies (always true), demonstrating the power of absorption in logical simplification.