Logical Connectives - Intermediate-Advanced Level: truth tables INTERMEDIATE-ADVANCED

Step 1: Understand 'only if' statements
'P only if Q' means 'If P, then Q' (P → Q)
This is DIFFERENT from 'If Q then P' (Q → P)

Key insight: 'only if' introduces a NECESSARY condition
Q is necessary for P (P cannot be true without Q)

Step 2: Identify components
Statement: The alarm rings only if there is an intruder
A: Alarm rings, I: There is an intruder

Step 3: Convert to logical form
Logical form: A → I
Equivalent: If the alarm rings, then there is an intruder

Step 4: Important distinction
Note: Intruder is necessary but not sufficient
'Only if' ≠ 'If and only if'
'Only if' gives one direction only (→)
'If and only if' gives both directions (↔)

Step 1: Understand necessary and sufficient conditions
• P is NECESSARY for Q: Q cannot be true without P (Q → P)
• P is SUFFICIENT for Q: P being true guarantees Q (P → Q)
• P is BOTH: P if and only if Q (P ↔ Q)

Step 2: Analyze the relationship
P: Being divisible by 4
Q: Being an even number

Step 3: Determine the condition type
All numbers divisible by 4 are even (sufficient), but not all even numbers are divisible by 4 (not necessary)

Answer: Sufficient but not necessary

Step 1: Understand the implication (→) operator
The implication p → q is False ONLY when p is True but q is False.
In all other cases, it is True.

Truth table for p → q:
p=T, q=T → Result=T
p=T, q=F → Result=F (the only False case)
p=F, q=T → Result=T
p=F, q=F → Result=T

Step 2: Apply the given values
p = True, q = False

Step 3: Evaluate p → q
Since p is True but q is False, the implication fails: p → q = False

Step 1: Understand the negation (NOT) operator
The negation ¬p simply reverses the truth value of p.
If p is True, then ¬p is False.
If p is False, then ¬p is True.

Step 2: Apply the given value
p = True

Step 3: Evaluate ¬p
Since p is True, ¬p = False
In other words: ¬p: It is not Monday is False

Step 1: Understand the biconditional (↔) operator
The biconditional p ↔ q is True when BOTH p and q have the SAME truth value.
It is False when p and q have DIFFERENT truth values.

Truth table for p ↔ q:
p=T, q=T → Result=T (same)
p=T, q=F → Result=F (different)
p=F, q=T → Result=F (different)
p=F, q=F → Result=T (same)

Step 2: Apply the given values
p = False, q = False

Step 3: Evaluate p ↔ q
Since p and q have the same truth value (False), p ↔ q = True

This is valid: If A ⊆ B and B ⊆ C, then A ⊆ C. All dogs (A) are mammals (B), all mammals (B) are animals (C), so all dogs (A) are animals (C).

Step 1: Understand the biconditional (↔) operator
The biconditional p ↔ q is True when BOTH p and q have the SAME truth value.
It is False when p and q have DIFFERENT truth values.

Truth table for p ↔ q:
p=T, q=T → Result=T (same)
p=T, q=F → Result=F (different)
p=F, q=T → Result=F (different)
p=F, q=F → Result=T (same)

Step 2: Apply the given values
p = True, q = True

Step 3: Evaluate p ↔ q
Since p and q have the same truth value (True), p ↔ q = True

Step 1: Understand the disjunction (OR) operator
The disjunction p ∨ q is True when AT LEAST ONE of p or q is True.
It is False ONLY when both p and q are False.

Step 2: Apply the given values
p = False, q = False

Step 3: Evaluate p ∨ q
Since both p and q are False, p ∨ q = False

Rule: Simplification (∧-elimination)

From a conjunction P ∧ Q, you can derive P (or Q) alone.

Answer: p

Step 1: Understand 'neither...nor' statements
'Neither p nor q' means 'not p AND not q'
Logical form: ¬p ∧ ¬q

Step 2: Build the truth table
¬p ∧ ¬q is True ONLY when both p and q are False
Truth table:
p=T, q=T → ¬T ∧ ¬T = F ∧ F = F
p=T, q=F → ¬T ∧ ¬F = F ∧ T = F
p=F, q=T → ¬F ∧ ¬T = T ∧ F = F
p=F, q=F → ¬F ∧ ¬F = T ∧ T = T

Step 3: Apply given values
p = T, q = F
¬p = F, ¬q = T
¬p ∧ ¬q = F

Answer: False

Step 1: Understand Exclusive OR (XOR)
Exclusive OR (p ⊕ q) is True when EXACTLY ONE of p or q is True.
It is False when both are True or both are False.

Truth table for p ⊕ q:
p=T, q=T → Result=F (both true)
p=T, q=F → Result=T (exactly one)
p=F, q=T → Result=T (exactly one)
p=F, q=F → Result=F (neither true)

Step 2: Apply the given values
p = True, q = True

Step 3: Evaluate the exclusive OR
Since both have the same truth value, the exclusive OR is False

Step 1: Understand the conjunction (AND) operator
The conjunction p ∧ q is True ONLY when BOTH p and q are True.

Step 2: Apply the given values
p = True, q = True

Step 3: Evaluate p ∧ q
Since both p and q are True, p ∧ q = True

'If P then Q' translates to P → Q (implication).

Answer: p → q

Inverse (¬p→¬q) is not equivalent to original. Contrapositive (¬q→¬p) is equivalent.

p=False, q=True: p→q=True, ¬p→¬q=False → Different!

Step 1: Understand the original statement
Original: p → q means "If it is raining, then the ground is wet"

Step 2: Understand Contrapositive
Contrapositive switches AND negates both parts: ¬q → ¬p
If the original is p → q, the contrapositive is ¬q → ¬p
Important: A conditional and its contrapositive are LOGICALLY EQUIVALENT

Step 3: Apply to our statement
Contrapositive: If the ground is wet is false, then it is raining is false

Step 1: Break down the expression
Expression: (p ∧ q) → r
Given: p=T, q=F, r=F

Step 2: Evaluate inner expressions first
p ∧ q = T ∧ F = F

Step 3: Evaluate outer expression
(F) → F = True
Remember: Implication is False only when antecedent is True and consequent is False

Disjoint means no element is in both, which is NOT (A AND B).

Answer: ¬(x ∈ A ∧ x ∈ B)

Step 1: Understand Exclusive OR (XOR)
Exclusive OR (p ⊕ q) is True when EXACTLY ONE of p or q is True.
It is False when both are True or both are False.

Truth table for p ⊕ q:
p=T, q=T → Result=F (both true)
p=T, q=F → Result=T (exactly one)
p=F, q=T → Result=T (exactly one)
p=F, q=F → Result=F (neither true)

Step 2: Apply the given values
p = False, q = False

Step 3: Evaluate the exclusive OR
Since both have the same truth value, the exclusive OR is False

Step 1: Break down the expression
Expression: (p ↔ q) ∧ r
Given: p=T, q=F, r=T

Step 2: Evaluate inner expressions first
p ↔ q = T ↔ F = F
(Biconditional is True when both have same value)

Step 3: Evaluate outer expression
F ∧ T = False
Remember: AND is True only when both operands are True

Step 1: Understand the original statement
Original: p → q means "If you study hard, then you will pass"

Step 2: Understand Contrapositive
Contrapositive switches AND negates both parts: ¬q → ¬p
If the original is p → q, the contrapositive is ¬q → ¬p
Important: A conditional and its contrapositive are LOGICALLY EQUIVALENT

Step 3: Apply to our statement
Contrapositive: If you will pass is false, then you study hard is false