Master Logical Equivalence - Intermediate-Advanced Level Problems Logical Equivalence INTERMEDIATE ADVANCED

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 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.

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.

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 conjunction equals the disjunction of negations.
These expressions ARE equivalent.

Verification with truth table:
p=T, q=T: ¬(T∧T)=F and ¬T∨¬T=F ✓
p=T, q=F: ¬(T∧F)=T and ¬T∨¬F=T ✓
p=F, q=T: ¬(F∧T)=T and ¬F∨¬T=T ✓
p=F, q=F: ¬(F∧F)=T and ¬F∨¬F=T ✓

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: q → p

Step 3: Test with truth table
p → q (implication) is NOT the same as q → p (converse).
Counter-example: p=F, q=T
p → q = F → T = T
q → p = T → F = F
Since they differ, they are NOT equivalent.

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 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.

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: q → p

Step 3: Test with truth table
p → q (implication) is NOT the same as q → p (converse).
Counter-example: p=F, q=T
p → q = F → T = T
q → p = T → F = F
Since they differ, they are NOT equivalent.

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 conjunction equals the disjunction of negations.
These expressions ARE equivalent.

Verification with truth table:
p=T, q=T: ¬(T∧T)=F and ¬T∨¬T=F ✓
p=T, q=F: ¬(T∧F)=T and ¬T∨¬F=T ✓
p=F, q=T: ¬(F∧T)=T and ¬F∨¬T=T ✓
p=F, q=F: ¬(F∧F)=T and ¬F∨¬F=T ✓

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.

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 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 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: q → p

Step 3: Test with truth table
p → q (implication) is NOT the same as q → p (converse).
Counter-example: p=F, q=T
p → q = F → T = T
q → p = T → F = F
Since they differ, they are NOT equivalent.

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 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.

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 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.

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 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 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.