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§ 2.2 Logically Equivalent Statements
Definition
DefinitionTwo expressions are logically equivalent provided that they have thesame truth value for all possible combinations of truth values for allvariables appearing in the two expressions. In this case, we writeX ≡ Y and say that X and Y are logically equivalent.
Are They Logically Equivalent?
ExampleIs p→ q ≡ p ∧ ¬q true?
How will we prove this?
Truth tables is a common way to go.
p q p→ q ¬q p ∧ ¬qT T T F FT F F T TF T T F FF F T T F
Conclusion? These statements are not logically equivalent.
Are They Logically Equivalent?
ExampleIs p→ q ≡ p ∧ ¬q true?
How will we prove this?
Truth tables is a common way to go.
p q p→ q ¬q p ∧ ¬qT T T F FT F F T TF T T F FF F T T F
Conclusion? These statements are not logically equivalent.
Are They Logically Equivalent?
ExampleIs p→ q ≡ p ∧ ¬q true?
How will we prove this?
Truth tables is a common way to go.
p q p→ q ¬q p ∧ ¬qT T T F FT F F T TF T T F FF F T T F
Conclusion? These statements are not logically equivalent.
Are They Logically Equivalent?
ExampleIs p→ q ≡ p ∧ ¬q true?
How will we prove this?
Truth tables is a common way to go.
p q p→ q ¬q p ∧ ¬q
T T T F FT F F T TF T T F FF F T T F
Conclusion? These statements are not logically equivalent.
Are They Logically Equivalent?
ExampleIs p→ q ≡ p ∧ ¬q true?
How will we prove this?
Truth tables is a common way to go.
p q p→ q ¬q p ∧ ¬qT T T F FT F F T TF T T F FF F T T F
Conclusion? These statements are not logically equivalent.
Are They Logically Equivalent?
ExampleIs p→ q ≡ p ∧ ¬q true?
How will we prove this?
Truth tables is a common way to go.
p q p→ q ¬q p ∧ ¬qT T T F FT F F T TF T T F FF F T T F
Conclusion?
These statements are not logically equivalent.
Are They Logically Equivalent?
ExampleIs p→ q ≡ p ∧ ¬q true?
How will we prove this?
Truth tables is a common way to go.
p q p→ q ¬q p ∧ ¬qT T T F FT F F T TF T T F FF F T T F
Conclusion? These statements are not logically equivalent.
Are They Logically Equivalent?
Example
Determine if (¬q)→ (p ∧ (¬p)) ≡ q.
p q ¬p ¬q p ∧ (¬p) (¬q)→ (p ∧ (¬p))T T F F F TT F F T F FF T T F F TF F T T F F
Conclusion? The statements are logically equivalent.
Are They Logically Equivalent?
Example
Determine if (¬q)→ (p ∧ (¬p)) ≡ q.
p q ¬p ¬q p ∧ (¬p) (¬q)→ (p ∧ (¬p))
T T F F F TT F F T F FF T T F F TF F T T F F
Conclusion? The statements are logically equivalent.
Are They Logically Equivalent?
Example
Determine if (¬q)→ (p ∧ (¬p)) ≡ q.
p q ¬p ¬q p ∧ (¬p) (¬q)→ (p ∧ (¬p))T T F F F TT F F T F FF T T F F TF F T T F F
Conclusion? The statements are logically equivalent.
Are They Logically Equivalent?
Example
Determine if (¬q)→ (p ∧ (¬p)) ≡ q.
p q ¬p ¬q p ∧ (¬p) (¬q)→ (p ∧ (¬p))T T F F F TT F F T F FF T T F F TF F T T F F
Conclusion?
The statements are logically equivalent.
Are They Logically Equivalent?
Example
Determine if (¬q)→ (p ∧ (¬p)) ≡ q.
p q ¬p ¬q p ∧ (¬p) (¬q)→ (p ∧ (¬p))T T F F F TT F F T F FF T T F F TF F T T F F
Conclusion? The statements are logically equivalent.
One More With 3 Statements
Example
Determine if p→ (q ∨ r) ≡ (¬q)→ ((¬p) ∨ r).
p q r q ∨ r p → (q ∨ r) ¬p ¬q (¬p) ∨ r (¬q) → ((¬p) ∨ r)T T T T T F F T TT T F T T F F F TT F T T T F T T TF T T T T T F T TT F F F F F T F FF T F T T T F T TF F T T T T T T TF F F F T T T T T
Therefore, the statements are logically equivalent.
One More With 3 Statements
Example
Determine if p→ (q ∨ r) ≡ (¬q)→ ((¬p) ∨ r).
p q r q ∨ r p → (q ∨ r) ¬p ¬q (¬p) ∨ r (¬q) → ((¬p) ∨ r)
T T T T T F F T TT T F T T F F F TT F T T T F T T TF T T T T T F T TT F F F F F T F FF T F T T T F T TF F T T T T T T TF F F F T T T T T
Therefore, the statements are logically equivalent.
One More With 3 Statements
Example
Determine if p→ (q ∨ r) ≡ (¬q)→ ((¬p) ∨ r).
p q r q ∨ r p → (q ∨ r) ¬p ¬q (¬p) ∨ r (¬q) → ((¬p) ∨ r)T T T T T F F T TT T F T T F F F TT F T T T F T T TF T T T T T F T TT F F F F F T F FF T F T T T F T TF F T T T T T T TF F F F T T T T T
Therefore, the statements are logically equivalent.
One More With 3 Statements
Example
Determine if p→ (q ∨ r) ≡ (¬q)→ ((¬p) ∨ r).
p q r q ∨ r p → (q ∨ r) ¬p ¬q (¬p) ∨ r (¬q) → ((¬p) ∨ r)T T T T T F F T TT T F T T F F F TT F T T T F T T TF T T T T T F T TT F F F F F T F FF T F T T T F T TF F T T T T T T TF F F F T T T T T
Therefore, the statements are logically equivalent.
DeMorgan’s Laws for Logic
TheoremDeMorgan’s LawsFor statements p and q,
The statement ¬(p ∧ q) is logically equivalent to ¬p ∨ ¬q. Thiscan be written as
¬(p ∧ q) ≡ ¬p ∨ ¬q
The statement ¬(p ∨ q) is logically equivalent to ¬p ∧ ¬q. Thiscan be written as
¬(p ∨ q) ≡ ¬p ∧ ¬q
Proof of DeMorgan’s Law
Generally, for proofs such as this, we use truth tables. So let’s not andsee how we could reason through a proof for the first case.
¬(p ∧ q) ≡ ¬p ∨ ¬q
Proof.p ∧ q is true only when p and q are both true statements. So, ¬(p ∧ q)is only false when p and q are both true statements.
On the other hand, the only way for a disjunction to be false is whenboth p and q are false. For ¬p ∨ ¬q, this only occurs when ¬p and ¬qare false, or when p and q are true.
Proof of DeMorgan’s Law
Generally, for proofs such as this, we use truth tables. So let’s not andsee how we could reason through a proof for the first case.
¬(p ∧ q) ≡ ¬p ∨ ¬q
Proof.p ∧ q is true only when p and q are both true statements.
So, ¬(p ∧ q)is only false when p and q are both true statements.
On the other hand, the only way for a disjunction to be false is whenboth p and q are false. For ¬p ∨ ¬q, this only occurs when ¬p and ¬qare false, or when p and q are true.
Proof of DeMorgan’s Law
Generally, for proofs such as this, we use truth tables. So let’s not andsee how we could reason through a proof for the first case.
¬(p ∧ q) ≡ ¬p ∨ ¬q
Proof.p ∧ q is true only when p and q are both true statements. So, ¬(p ∧ q)is only false when p and q are both true statements.
On the other hand, the only way for a disjunction to be false is whenboth p and q are false. For ¬p ∨ ¬q, this only occurs when ¬p and ¬qare false, or when p and q are true.
Proof of DeMorgan’s Law
Generally, for proofs such as this, we use truth tables. So let’s not andsee how we could reason through a proof for the first case.
¬(p ∧ q) ≡ ¬p ∨ ¬q
Proof.p ∧ q is true only when p and q are both true statements. So, ¬(p ∧ q)is only false when p and q are both true statements.
On the other hand, the only way for a disjunction to be false is whenboth p and q are false.
For ¬p ∨ ¬q, this only occurs when ¬p and ¬qare false, or when p and q are true.
Proof of DeMorgan’s Law
Generally, for proofs such as this, we use truth tables. So let’s not andsee how we could reason through a proof for the first case.
¬(p ∧ q) ≡ ¬p ∨ ¬q
Proof.p ∧ q is true only when p and q are both true statements. So, ¬(p ∧ q)is only false when p and q are both true statements.
On the other hand, the only way for a disjunction to be false is whenboth p and q are false. For ¬p ∨ ¬q, this only occurs when ¬p and ¬qare false, or when p and q are true.
Logical Equivalence and Conditional Statements
TheoremFor statements p and q,
1 The conditional statement p→ q is logically equivalent to¬p ∨ q.
2 The statement ¬(p→ q) is logically equivalent to p ∧ ¬q.3 The conditional statement p→ q is logically equivalent to its
contrapositive ¬q→ ¬p.
This is a theorem in the book but it is not proved, so we will do sonow with truth tables.
Logical Equivalence and Conditional Statements
TheoremFor statements p and q,
1 The conditional statement p→ q is logically equivalent to¬p ∨ q.
2 The statement ¬(p→ q) is logically equivalent to p ∧ ¬q.
3 The conditional statement p→ q is logically equivalent to itscontrapositive ¬q→ ¬p.
This is a theorem in the book but it is not proved, so we will do sonow with truth tables.
Logical Equivalence and Conditional Statements
TheoremFor statements p and q,
1 The conditional statement p→ q is logically equivalent to¬p ∨ q.
2 The statement ¬(p→ q) is logically equivalent to p ∧ ¬q.3 The conditional statement p→ q is logically equivalent to its
contrapositive ¬q→ ¬p.
This is a theorem in the book but it is not proved, so we will do sonow with truth tables.
Logical Equivalence and Conditional Statements
TheoremFor statements p and q,
1 The conditional statement p→ q is logically equivalent to¬p ∨ q.
2 The statement ¬(p→ q) is logically equivalent to p ∧ ¬q.3 The conditional statement p→ q is logically equivalent to its
contrapositive ¬q→ ¬p.
This is a theorem in the book but it is not proved, so we will do sonow with truth tables.
Proof of Part 1
We will show p→ q ≡ ¬p ∨ q.
p q p→ q ¬p ¬p ∨ qT T T F TT F F F FF T T T TF F T T T
Proof of Part 1
We will show p→ q ≡ ¬p ∨ q.
p q p→ q ¬p ¬p ∨ qT T T F TT F F F FF T T T TF F T T T
Proof of Part 2
We will show ¬(p→ q) ≡ p ∧ ¬q
p q p→ q ¬(p→ q) ¬q p ∧ ¬qT T T F F FT F F T T TF T T F F FF F T F T F
Proof of Part 2
We will show ¬(p→ q) ≡ p ∧ ¬q
p q p→ q ¬(p→ q) ¬q p ∧ ¬qT T T F F FT F F T T TF T T F F FF F T F T F
Proof of Part 3
We will show p→ q ≡ ¬q→ ¬p
p q p→ q ¬q ¬p ¬q→ ¬pT T T F F TT F F T F FF T T F T TF F T T T T
Proof of Part 3
We will show p→ q ≡ ¬q→ ¬p
p q p→ q ¬q ¬p ¬q→ ¬pT T T F F TT F F T F FF T T F T TF F T T T T
Negation
Negation of a conditional statement is not another conditionalstatement.
Example
¬(p→ q) ≡ p ∧ ¬q
We will verify this using truth tables.
p q p→ q ¬(→ q) ¬q p ∧ ¬qT T T F F FT F F T T TF T T F F FF F T F T F
Negation
Negation of a conditional statement is not another conditionalstatement.
Example
¬(p→ q) ≡ p ∧ ¬q
We will verify this using truth tables.
p q p→ q ¬(→ q) ¬q p ∧ ¬qT T T F F FT F F T T TF T T F F FF F T F T F
Negation
Negation of a conditional statement is not another conditionalstatement.
Example
¬(p→ q) ≡ p ∧ ¬q
We will verify this using truth tables.
p q p→ q ¬(→ q) ¬q p ∧ ¬qT T T F F FT F F T T TF T T F F FF F T F T F
Negation
Negation of a conditional statement is not another conditionalstatement.
Example
¬(p→ q) ≡ p ∧ ¬q
We will verify this using truth tables.
p q p→ q ¬(→ q) ¬q p ∧ ¬qT T T F F FT F F T T TF T T F F FF F T F T F
Important Equivalences
De Morgan’s Laws ¬(p ∧ q) ≡ ¬p ∨ ¬q¬(p ∨ q) ≡ ¬p ∧ ¬q
Conditional Statements p→ q ≡ ¬q→ ¬pp→ q ≡ ¬p ∨ q¬(p→ q) ≡ p ∧ ¬q
Biconditional Statement (p↔ q) ≡ (p→ q) ∧ (q→ p)Double Negation ¬(¬p) ≡ pDistributive Laws p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)Conditionals with p→ (q ∨ r) ≡ (p ∧ ¬q)→ rDisjunctions (p ∨ q)→ r ≡ (p→ r) ∧ (q→ r)
These are truth table verifications, and a couple are homeworkexercises.
Important Equivalences
De Morgan’s Laws ¬(p ∧ q) ≡ ¬p ∨ ¬q¬(p ∨ q) ≡ ¬p ∧ ¬q
Conditional Statements p→ q ≡ ¬q→ ¬pp→ q ≡ ¬p ∨ q¬(p→ q) ≡ p ∧ ¬q
Biconditional Statement (p↔ q) ≡ (p→ q) ∧ (q→ p)Double Negation ¬(¬p) ≡ pDistributive Laws p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)Conditionals with p→ (q ∨ r) ≡ (p ∧ ¬q)→ rDisjunctions (p ∨ q)→ r ≡ (p→ r) ∧ (q→ r)
These are truth table verifications, and a couple are homeworkexercises.
Important Equivalences
De Morgan’s Laws ¬(p ∧ q) ≡ ¬p ∨ ¬q¬(p ∨ q) ≡ ¬p ∧ ¬q
Conditional Statements p→ q ≡ ¬q→ ¬pp→ q ≡ ¬p ∨ q¬(p→ q) ≡ p ∧ ¬q
Biconditional Statement (p↔ q) ≡ (p→ q) ∧ (q→ p)
Double Negation ¬(¬p) ≡ pDistributive Laws p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)Conditionals with p→ (q ∨ r) ≡ (p ∧ ¬q)→ rDisjunctions (p ∨ q)→ r ≡ (p→ r) ∧ (q→ r)
These are truth table verifications, and a couple are homeworkexercises.
Important Equivalences
De Morgan’s Laws ¬(p ∧ q) ≡ ¬p ∨ ¬q¬(p ∨ q) ≡ ¬p ∧ ¬q
Conditional Statements p→ q ≡ ¬q→ ¬pp→ q ≡ ¬p ∨ q¬(p→ q) ≡ p ∧ ¬q
Biconditional Statement (p↔ q) ≡ (p→ q) ∧ (q→ p)Double Negation ¬(¬p) ≡ pDistributive Laws p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
Conditionals with p→ (q ∨ r) ≡ (p ∧ ¬q)→ rDisjunctions (p ∨ q)→ r ≡ (p→ r) ∧ (q→ r)
These are truth table verifications, and a couple are homeworkexercises.
Important Equivalences
De Morgan’s Laws ¬(p ∧ q) ≡ ¬p ∨ ¬q¬(p ∨ q) ≡ ¬p ∧ ¬q
Conditional Statements p→ q ≡ ¬q→ ¬pp→ q ≡ ¬p ∨ q¬(p→ q) ≡ p ∧ ¬q
Biconditional Statement (p↔ q) ≡ (p→ q) ∧ (q→ p)Double Negation ¬(¬p) ≡ pDistributive Laws p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)Conditionals with p→ (q ∨ r) ≡ (p ∧ ¬q)→ r
Disjunctions (p ∨ q)→ r ≡ (p→ r) ∧ (q→ r)
These are truth table verifications, and a couple are homeworkexercises.
Important Equivalences
De Morgan’s Laws ¬(p ∧ q) ≡ ¬p ∨ ¬q¬(p ∨ q) ≡ ¬p ∧ ¬q
Conditional Statements p→ q ≡ ¬q→ ¬pp→ q ≡ ¬p ∨ q¬(p→ q) ≡ p ∧ ¬q
Biconditional Statement (p↔ q) ≡ (p→ q) ∧ (q→ p)Double Negation ¬(¬p) ≡ pDistributive Laws p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)Conditionals with p→ (q ∨ r) ≡ (p ∧ ¬q)→ rDisjunctions (p ∨ q)→ r ≡ (p→ r) ∧ (q→ r)
These are truth table verifications, and a couple are homeworkexercises.
Important Equivalences
De Morgan’s Laws ¬(p ∧ q) ≡ ¬p ∨ ¬q¬(p ∨ q) ≡ ¬p ∧ ¬q
Conditional Statements p→ q ≡ ¬q→ ¬pp→ q ≡ ¬p ∨ q¬(p→ q) ≡ p ∧ ¬q
Biconditional Statement (p↔ q) ≡ (p→ q) ∧ (q→ p)Double Negation ¬(¬p) ≡ pDistributive Laws p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)Conditionals with p→ (q ∨ r) ≡ (p ∧ ¬q)→ rDisjunctions (p ∨ q)→ r ≡ (p→ r) ∧ (q→ r)
These are truth table verifications, and a couple are homeworkexercises.
A Final Proof
Let p and q be statements. show that
((p ∨ q) ∧ ¬(p ∧ q)) ≡ ¬(p↔ q)
Proof
(p ∨ q) ∧ ¬(p ∧ q) Given statement(p ∨ q) ∧ (¬p ∨ ¬q) DeMorgan’s Law(p ∧ (¬p ∨ ¬q)) ∨ (q ∧ (¬p ∨ ¬q)) Distribution Law(p ∧ ¬p) ∨ (p ∧ ¬q) ∨ (q ∧ ¬p) ∨ (q ∧ ¬q) Distribution Law(p ∧ ¬q) ∨ (q ∧ ¬p) Impossibility¬(p→ q) ∨ ¬(q→ p) Conditional Law¬((p→ q) ∧ (q→ p)) DeMorgan’s Law¬(p↔ q) Definition of IFF
A Final Proof
Let p and q be statements. show that
((p ∨ q) ∧ ¬(p ∧ q)) ≡ ¬(p↔ q)
Proof
(p ∨ q) ∧ ¬(p ∧ q) Given statement
(p ∨ q) ∧ (¬p ∨ ¬q) DeMorgan’s Law(p ∧ (¬p ∨ ¬q)) ∨ (q ∧ (¬p ∨ ¬q)) Distribution Law(p ∧ ¬p) ∨ (p ∧ ¬q) ∨ (q ∧ ¬p) ∨ (q ∧ ¬q) Distribution Law(p ∧ ¬q) ∨ (q ∧ ¬p) Impossibility¬(p→ q) ∨ ¬(q→ p) Conditional Law¬((p→ q) ∧ (q→ p)) DeMorgan’s Law¬(p↔ q) Definition of IFF
A Final Proof
Let p and q be statements. show that
((p ∨ q) ∧ ¬(p ∧ q)) ≡ ¬(p↔ q)
Proof
(p ∨ q) ∧ ¬(p ∧ q) Given statement(p ∨ q) ∧ (¬p ∨ ¬q) DeMorgan’s Law
(p ∧ (¬p ∨ ¬q)) ∨ (q ∧ (¬p ∨ ¬q)) Distribution Law(p ∧ ¬p) ∨ (p ∧ ¬q) ∨ (q ∧ ¬p) ∨ (q ∧ ¬q) Distribution Law(p ∧ ¬q) ∨ (q ∧ ¬p) Impossibility¬(p→ q) ∨ ¬(q→ p) Conditional Law¬((p→ q) ∧ (q→ p)) DeMorgan’s Law¬(p↔ q) Definition of IFF
A Final Proof
Let p and q be statements. show that
((p ∨ q) ∧ ¬(p ∧ q)) ≡ ¬(p↔ q)
Proof
(p ∨ q) ∧ ¬(p ∧ q) Given statement(p ∨ q) ∧ (¬p ∨ ¬q) DeMorgan’s Law(p ∧ (¬p ∨ ¬q)) ∨ (q ∧ (¬p ∨ ¬q)) Distribution Law
(p ∧ ¬p) ∨ (p ∧ ¬q) ∨ (q ∧ ¬p) ∨ (q ∧ ¬q) Distribution Law(p ∧ ¬q) ∨ (q ∧ ¬p) Impossibility¬(p→ q) ∨ ¬(q→ p) Conditional Law¬((p→ q) ∧ (q→ p)) DeMorgan’s Law¬(p↔ q) Definition of IFF
A Final Proof
Let p and q be statements. show that
((p ∨ q) ∧ ¬(p ∧ q)) ≡ ¬(p↔ q)
Proof
(p ∨ q) ∧ ¬(p ∧ q) Given statement(p ∨ q) ∧ (¬p ∨ ¬q) DeMorgan’s Law(p ∧ (¬p ∨ ¬q)) ∨ (q ∧ (¬p ∨ ¬q)) Distribution Law(p ∧ ¬p) ∨ (p ∧ ¬q) ∨ (q ∧ ¬p) ∨ (q ∧ ¬q) Distribution Law
(p ∧ ¬q) ∨ (q ∧ ¬p) Impossibility¬(p→ q) ∨ ¬(q→ p) Conditional Law¬((p→ q) ∧ (q→ p)) DeMorgan’s Law¬(p↔ q) Definition of IFF
A Final Proof
Let p and q be statements. show that
((p ∨ q) ∧ ¬(p ∧ q)) ≡ ¬(p↔ q)
Proof
(p ∨ q) ∧ ¬(p ∧ q) Given statement(p ∨ q) ∧ (¬p ∨ ¬q) DeMorgan’s Law(p ∧ (¬p ∨ ¬q)) ∨ (q ∧ (¬p ∨ ¬q)) Distribution Law(p ∧ ¬p) ∨ (p ∧ ¬q) ∨ (q ∧ ¬p) ∨ (q ∧ ¬q) Distribution Law(p ∧ ¬q) ∨ (q ∧ ¬p) Impossibility
¬(p→ q) ∨ ¬(q→ p) Conditional Law¬((p→ q) ∧ (q→ p)) DeMorgan’s Law¬(p↔ q) Definition of IFF
A Final Proof
Let p and q be statements. show that
((p ∨ q) ∧ ¬(p ∧ q)) ≡ ¬(p↔ q)
Proof
(p ∨ q) ∧ ¬(p ∧ q) Given statement(p ∨ q) ∧ (¬p ∨ ¬q) DeMorgan’s Law(p ∧ (¬p ∨ ¬q)) ∨ (q ∧ (¬p ∨ ¬q)) Distribution Law(p ∧ ¬p) ∨ (p ∧ ¬q) ∨ (q ∧ ¬p) ∨ (q ∧ ¬q) Distribution Law(p ∧ ¬q) ∨ (q ∧ ¬p) Impossibility¬(p→ q) ∨ ¬(q→ p) Conditional Law
¬((p→ q) ∧ (q→ p)) DeMorgan’s Law¬(p↔ q) Definition of IFF
A Final Proof
Let p and q be statements. show that
((p ∨ q) ∧ ¬(p ∧ q)) ≡ ¬(p↔ q)
Proof
(p ∨ q) ∧ ¬(p ∧ q) Given statement(p ∨ q) ∧ (¬p ∨ ¬q) DeMorgan’s Law(p ∧ (¬p ∨ ¬q)) ∨ (q ∧ (¬p ∨ ¬q)) Distribution Law(p ∧ ¬p) ∨ (p ∧ ¬q) ∨ (q ∧ ¬p) ∨ (q ∧ ¬q) Distribution Law(p ∧ ¬q) ∨ (q ∧ ¬p) Impossibility¬(p→ q) ∨ ¬(q→ p) Conditional Law¬((p→ q) ∧ (q→ p)) DeMorgan’s Law
¬(p↔ q) Definition of IFF
A Final Proof
Let p and q be statements. show that
((p ∨ q) ∧ ¬(p ∧ q)) ≡ ¬(p↔ q)
Proof
(p ∨ q) ∧ ¬(p ∧ q) Given statement(p ∨ q) ∧ (¬p ∨ ¬q) DeMorgan’s Law(p ∧ (¬p ∨ ¬q)) ∨ (q ∧ (¬p ∨ ¬q)) Distribution Law(p ∧ ¬p) ∨ (p ∧ ¬q) ∨ (q ∧ ¬p) ∨ (q ∧ ¬q) Distribution Law(p ∧ ¬q) ∨ (q ∧ ¬p) Impossibility¬(p→ q) ∨ ¬(q→ p) Conditional Law¬((p→ q) ∧ (q→ p)) DeMorgan’s Law¬(p↔ q) Definition of IFF
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