Discrete Mathematics - Propositional Logic

The rules of mathematical logic specify methods of reasoning mathematical statements. Greek philosopher, Aristotle, was the pioneer of logical reasoning. Logical reasoning provides the theoretical base for many areas of mathematics and consequently computer science. It has many practical applications in computer science like design of computing machines, artificial intelligence, definition of data structures for programming languages etc.

Propositional Logic is concerned with statements to which the truth values, “true” and “false”, can be assigned. The purpose is to analyze these statements either individually or in a composite manner.

Prepositional Logic – Definition

A proposition is a collection of declarative statements that has either a truth value "true” or a truth value "false". A propositional consists of propositional variables and connectives. We denote the propositional variables by capital letters (A, B, etc). The connectives connect the propositional variables.

Some examples of Propositions are given below −

• "Man is Mortal", it returns truth value “TRUE”
• "12 + 9 = 3 – 2", it returns truth value “FALSE”

The following is not a Proposition −

• "A is less than 2". It is because unless we give a specific value of A, we cannot say whether the statement is true or false.

Connectives

In propositional logic generally we use five connectives which are −

• OR (∨$\vee$)
• AND (∧$\wedge$)
• Negation/ NOT (¬$\mathrm{\neg }$)
• Implication / if-then (→$\to$)
• If and only if (⇔$\iff$).

OR (∨$\vee$) − The OR operation of two propositions A and B (written as A∨B$A\vee B$) is true if at least any of the propositional variable A or B is true.

The truth table is as follows −

A	B	A ∨ B
True	True	True
True	False	True
False	True	True
False	False	False

AND (∧$\wedge$) − The AND operation of two propositions A and B (written as A∧B$A\wedge B$) is true if both the propositional variable A and B is true.

The truth table is as follows −

A	B	A ∧ B
True	True	True
True	False	False
False	True	False
False	False	False

Negation (¬$\mathrm{\neg }$) − The negation of a proposition A (written as ¬A$\mathrm{\neg }A$) is false when A is true and is true when A is false.

The truth table is as follows −

A	¬ A
True	False
False	True

Implication / if-then (→$\to$) − An implication A→B$A\to B$ is the proposition “if A, then B”. It is false if A is true and B is false. The rest cases are true.

The truth table is as follows −

A	B	A → B
True	True	True
True	False	False
False	True	True
False	False	True

If and only if (⇔$\iff$) − A⇔B$A\iff B$ is bi-conditional logical connective which is true when p and q are same, i.e. both are false or both are true.

The truth table is as follows −

A	B	A ⇔ B
True	True	True
True	False	False
False	True	False
False	False	True

Tautologies

A Tautology is a formula which is always true for every value of its propositional variables.

Example − Prove [(A→B)∧A]→B$[(A\to B)\wedge A]\to B$ is a tautology

The truth table is as follows −

A	B	A → B	(A → B) ∧ A	[( A → B ) ∧ A] → B
True	True	True	True	True
True	False	False	False	True
False	True	True	False	True
False	False	True	False	True

As we can see every value of [(A→B)∧A]→B$[(A\to B)\wedge A]\to B$ is "True", it is a tautology.

Contradictions

A Contradiction is a formula which is always false for every value of its propositional variables.

Example − Prove (A∨B)∧[(¬A)∧(¬B)]$(A\vee B)\wedge [(\mathrm{\neg }A)\wedge (\mathrm{\neg }B)]$ is a contradiction

The truth table is as follows −

A	B	A ∨ B	¬ A	¬ B	(¬ A) ∧ ( ¬ B)	(A ∨ B) ∧ [( ¬ A) ∧ (¬ B)]
True	True	True	False	False	False	False
True	False	True	False	True	False	False
False	True	True	True	False	False	False
False	False	False	True	True	True	False

As we can see every value of (A∨B)∧[(¬A)∧(¬B)]$(A\vee B)\wedge [(\mathrm{\neg }A)\wedge (\mathrm{\neg }B)]$ is “False”, it is a contradiction.

Contingency

A Contingency is a formula which has both some true and some false values for every value of its propositional variables.

Example − Prove (A∨B)∧(¬A)$(A\vee B)\wedge (\mathrm{\neg }A)$ a contingency

The truth table is as follows −

A	B	A ∨ B	¬ A	(A ∨ B) ∧ (¬ A)
True	True	True	False	False
True	False	True	False	False
False	True	True	True	True
False	False	False	True	False

As we can see every value of (A∨B)∧(¬A)$(A\vee B)\wedge (\mathrm{\neg }A)$ has both “True” and “False”, it is a contingency.