Logical Connectives - Advanced Level: boolean logic 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 '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 = T
¬p = T, ¬q = F
¬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 = False

Step 3: Evaluate the exclusive OR
Since exactly one of p or q is True, the exclusive OR is True

'Either P or Q, but not both' is exclusive OR: P ⊕ Q.

Answer: p ⊕ q

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 '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 → q)

Step 3: Test all possible combinations
This simplifies to q ∨ ¬q, which is always True

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

If A knave → statement 'at least one knave' true → but knave can't tell truth → impossible. So A knight → statement true → at least one knave → since A knight, B must be knave.

Step 1: Test if A is a knight
If A is a knight, A tells the truth.
But A says 'I am a knave', which would be a lie.
Contradiction! A cannot be a knight.

Step 2: Test if A is a knave
If A is a knave, A lies.
But A says 'I am a knave', which would be true.
Contradiction! A cannot be a knave.

Step 3: Conclusion
Neither possibility works.
This statement is a LOGICAL PARADOX.
No one can truthfully or falsely claim to be a knave.

Answer: This statement is impossible

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 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)
Expression 2: p

Step 3: Apply Double Negation law
Two negations cancel each other out.
¬(¬p) simply gives back p.
These expressions ARE equivalent.

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 = 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 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 = True

Step 3: Evaluate the exclusive OR
Since exactly one of p or q is True, the exclusive OR is True

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 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 = T
p=F: F ∨ T = T
Result: Always True → TAUTOLOGY
This is the Law of Excluded Middle

Implication is not symmetric. Converse is not equivalent to original.

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

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

Union means element is in AT LEAST ONE set, which is logical OR.

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