Logical Connectives - Advanced Level: logical gates ADVANCED

Inclusive OR is true when both are true; exclusive OR is false when both are true.

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

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
A statement cannot be both true and 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 De Morgan's Law
De Morgan's Law states: ¬(p ∨ q) ≡ ¬p ∧ ¬q
The negation of a disjunction equals the conjunction of negations.
These expressions ARE equivalent.

This is valid in form, but the premise 'All birds can fly' is false. Validity is about logical structure, not factual truth. Form: All A are B, C is A → C is B.

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

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 it is raining is false, then the ground is wet 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 = True, q = True

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

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 ∧ q)
Expression 2: ¬p ∧ ¬q

Step 3: Test with truth table
Testing: ¬(p ∧ q) vs ¬p ∧ ¬q
Counter-example: p=T, q=F
¬(T ∧ F) = ¬F = T
¬T ∧ ¬F = F ∧ T = F
Since they differ, they are NOT equivalent.

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: Being a triangle
Q: Having three sides

Step 3: Determine the condition type
A shape is a triangle if and only if it has three sides

Answer: Necessary and sufficient

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

Rule: Simplification (∧-elimination)

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

Answer: p

Argument form: Modus Ponens

This is Modus Ponens: If P → Q and P are true, then Q must be true.

Conclusion: This argument is VALID.

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

Rule: Conjunction Introduction (∧-intro)

If you have P and you have Q, you can combine them into P ∧ Q.

Answer: p ∧ 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: Being a square
Q: Being a rectangle

Step 3: Determine the condition type
All squares are rectangles (sufficient), but not all rectangles are squares (not necessary)

Answer: Sufficient but not necessary

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 = False, q = False

Step 3: Evaluate p ∧ q
Since at least one of p or q is False, p ∧ q = False

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

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

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