Logical Connectives - Intermediate Level: conditional statements INTERMEDIATE

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

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 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 = False

Step 3: Evaluate ¬p
Since p is False, ¬p = True
In other words: ¬p: The door is not locked is True

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: The door is not locked is False

Step 1: Understand what logical equivalence means
Two expressions are logically equivalent if they have the same truth value for ALL possible combinations of variables.

Step 2: Analyze the expressions
Expression 1: p → q
Expression 2: ¬p ∨ q

Step 3: Apply Implication equivalence
The implication p → q is equivalent to ¬p ∨ q
This is a fundamental law in logic.
These expressions ARE equivalent.

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: Analyze A's statement
A says: 'We are both knaves'

Step 2: Test if A is a knight
If A is a knight, then A tells the truth.
But then 'We are both knaves' would be true.
This means A is a knave, which contradicts our assumption.
Therefore, A cannot be a knight.

Step 3: Test if A is a knave
If A is a knave, then A lies.
A's statement 'We are both knaves' must be false.
For 'both knaves' to be false, at least one must be a knight.
Since A is a knave, B must be a knight.

Step 4: Verify
A (knave) lies: 'We are both knaves' is indeed false ✓
B is a knight ✓

Answer: A is a knave, B is a knight

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

Answer: ¬(x ∈ A ∧ x ∈ B)

'P and Q' translates to P ∧ Q (conjunction).

Answer: p ∧ q

Step 1: Break down the compound expression
Expression: (p ∧ q) ∨ r

Step 2: Evaluate inner expression first
p ∧ q = True ∧ False = False

Step 3: Evaluate outer expression
(False) ∨ False = False
Since OR is True when at least one operand is True

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 = False, q = True

Step 3: Evaluate p → q
Since this is not the case where p is True and q is False, p → q = True

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

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 you study hard is false, then you will pass 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: 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

Step 1: Understand 'unless' statements
'P unless Q' means 'If not Q, then P'
In logical form: 'P unless Q' ≡ '¬Q → P'

Step 2: Identify components
Original: You will fail unless you study
p: You will fail, q: You study

Step 3: Convert to if-then form
'unless' tells us what happens if the condition is NOT met
Logical form: ¬q → p

Step 4: Write in English
Equivalent statement: If you do not study, then you will fail

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 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 = False, q = True

Step 3: Evaluate p → q
Since this is not the case where p is True and q is False, p → q = True

Step 1: Understand 'unless' statements
'P unless Q' means 'If not Q, then P'
In logical form: 'P unless Q' ≡ '¬Q → P'

Step 2: Identify components
Original: The plant will die unless you water it
p: The plant will die, q: You water it

Step 3: Convert to if-then form
'unless' tells us what happens if the condition is NOT met
Logical form: ¬q → p

Step 4: Write in English
Equivalent statement: If you do not water it, then the plant will die

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 = F, q = F
¬p = T, ¬q = T
¬p ∧ ¬q = T

Answer: True

Test cases:
- A knight → statement true: 'knight or B knave' → true (first part true) ✓. B says 'A knave' which is false, so B knave ✓.
- A knave → statement false: 'knight or B knave' false → both parts false → 'knight' false (ok), 'B knave' false → B knight. Then B says 'A knave' which would be true (since A knave), but B knight must tell truth ✓. This also works? Wait, if A knave and B knight: A's statement 'knight or B knave' = 'false or false' = false (knave lies) ✓. B's statement 'A knave' = true (knight truth) ✓. Two solutions? But puzzle assumes unique. Check carefully: With A knave, B knight: A says 'knight or B knave' = 'false or false' = false (good lie). B says 'A knave' = true (good truth). Both work. So puzzle ambiguous. We'll take first solution: A knight, B knave.

Step 1: Understand 'unless' statements
'P unless Q' means 'If not Q, then P'
In logical form: 'P unless Q' ≡ '¬Q → P'

Step 2: Identify components
Original: She won't come unless you invite her
p: She won't come, q: You invite her

Step 3: Convert to if-then form
'unless' tells us what happens if the condition is NOT met
Logical form: ¬q → p

Step 4: Write in English
Equivalent statement: If you do not invite her, then she won't come