Logical Connectives - Beginner-Intermediate Level: implication BEGINNER-INTERMEDIATE

Step 1: Understand the classifications
• Tautology: Always True for all possible truth values
• Contradiction: Always False for all possible truth values
• Contingent: True for some values, False for others

Step 2: Analyze the expression
Expression: p ∧ ¬p

Step 3: Test all possible combinations
Truth table:
p=T: T ∧ F = F
p=F: F ∧ T = F
Result: Always False → CONTRADICTION
This violates the Law of Non-Contradiction

Step 1: Understand the original statement
Original: p → q means "If a number is divisible by 4, then it is even"

Step 2: Understand Converse
Converse switches the hypothesis and conclusion: q → p
If the original is p → q, the converse is q → p

Step 3: Apply to our statement
Converse: If it is even, then a number is divisible by 4

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 '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: A number is prime only if it is greater than 1
P: Number is prime, G: Greater than 1

Step 3: Convert to logical form
Logical form: P → G
Equivalent: If a number is prime, then it is greater than 1

Step 4: Important distinction
Note: Being greater than 1 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: Having oxygen
Q: Fire burning

Step 3: Determine the condition type
Fire needs oxygen (necessary), but oxygen alone doesn't guarantee fire (not sufficient)

Answer: Necessary but not sufficient

Case analysis: If A knight → B knave (A's truth) → A and C different (B's lie) → C knight? But C says 'A is knight' which would be true, consistent. Wait, need full check.

Actually solve: Assume A knight → 'B knave' true → B knave → B's statement 'A and C same' is false → A and C different → C knave → C says 'A knight' which is false (since A knight?) Contradiction.

Therefore A knave → 'B knave' false → B knight → B's statement true → A and C same → C knave → C says 'A knight' false (since A knave) ✓. Solution: A knave, B knight, C knave.

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

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: You will pass only if you study
P: You pass, S: You study

Step 3: Convert to logical form
Logical form: P → S
Equivalent: If you pass, then you studied

Step 4: Important distinction
Note: Studying 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 the classifications
• Tautology: Always True for all possible truth values
• Contradiction: Always False for all possible truth values
• Contingent: True for some values, False for others

Step 2: Analyze the expression
Expression: (p → q) → p

Step 3: Test all possible combinations
This is contingent - depends on values of p and q

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: Studying
Q: Passing the exam

Step 3: Determine the condition type
You need to study to pass (necessary), but studying alone doesn't guarantee passing (not sufficient)

Answer: Necessary but not sufficient

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 (↔)

Case analysis: If A knight → B knave (A's truth) → A and C different (B's lie) → C knight? But C says 'A is knight' which would be true, consistent. Wait, need full check.

Actually solve: Assume A knight → 'B knave' true → B knave → B's statement 'A and C same' is false → A and C different → C knave → C says 'A knight' which is false (since A knight?) Contradiction.

Therefore A knave → 'B knave' false → B knight → B's statement true → A and C same → C knave → C says 'A knight' false (since A knave) ✓. Solution: A knave, B knight, C knave.

Subset means IF element is in A, THEN it must be in B, which is implication.

Answer: x ∈ A → x ∈ B

'Unless P, then Q' means ¬P → Q. If you don't study, then you fail.

Answer: ¬s → f

Step 1: Understand the original statement
Original: p → q means "If a number is divisible by 4, then it is even"

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 it is even is false, then a number is divisible by 4 is false

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 original statement
Original: p → q means "If the alarm rings, then I wake up"

Step 2: Understand Inverse
Inverse negates both parts: ¬p → ¬q
If the original is p → q, the inverse is ¬p → ¬q

Step 3: Apply to our statement
Inverse: If the alarm rings is false, then I wake up is false

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 '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

Tradition doesn't justify correctness or optimality.