Tag: Negation

Logical Connectives | Truth Tables | Examples

Propositional Logic

Propositions-

 

Before you go through this article, make sure that you have gone through the previous article on Propositions.

 

We have discussed-

  • Proposition is a declarative statement that is either true or false but not both.
  • Connectives are used to combine the propositions.

 

 

In this article, we will discuss about connectives in propositional logic.

 

Logical Connectives-

 

Connectives are the operators that are used to combine one or more propositions.

 

In propositional logic, there are 5 basic connectives-

 

Name of Connective Connective Word Symbol
Negation Not ⌉ or ∼ or ‘ or –
Conjunction And ∧
Disjunction Or ∨
Conditional If-then →
Biconditional If and only if ↔

 

1. Negation-

 

If p is a proposition, then negation of p is a proposition which is-

  • True when p is false
  • False when p is true.

 

Truth Table-

 

p ∼p
F T
T F

 

Example-

 

If p : It is raining outside.

Then, Negation of p is-

∼p : It is not raining outside.

 

2. Conjunction-

 

If p and q are two propositions, then conjunction of p and q is a proposition which is-

  • True when both p and q are true
  • False when both p and q are false

 

Truth Table-

 

p q p ∧ q
F F F
F T F
T F F
T T T

 

Example-

 

If p and q are two propositions where-

  • p : 2 + 4 = 6
  • q : It is raining outside.

 

Then, conjunction of p and q is-

p ∧ q : 2 + 4 = 6 and it is raining outside

 

3. Disjunction-

 

If p and q are two propositions, then disjunction of p and q is a proposition which is-

  • True when either one of p or q or both are true
  • False when both p and q are false

 

Truth Table-

 

p q p ∨ q
F F F
F T T
T F T
T T T

 

Example-

 

If p and q are two propositions where-

  • p : 2 + 4 = 6
  • q : It is raining outside

 

Then, disjunction of p and q is-

p ∨ q : 2 + 4 = 6 or it is raining outside

 

4. Conditional-

 

If p and q are two propositions, then-

  • Proposition of the type “If p then q” is called a conditional or implication proposition.
  • It is true when both p and q are true or when p is false.
  • It is false when p is true and q is false.

 

Truth Table-

 

p q p → q
F F T
F T T
T F F
T T T

 

Examples-

 

  • If a = b and b = c then a = c.
  • If I will go to Australia, then I will earn more money.

 

5. Biconditional-

 

If p and q are two propositions, then-

  • Proposition of the type “p if and only if q” is called a biconditional or bi-implication proposition.
  • It is true when either both p and q are true or both p and q are false.
  • It is false in all other cases.

 

Truth Table-

 

p q p ↔ q
F F T
F T F
T F F
T T T

 

Examples-

 

  • He goes to play a match if and only if it does not rain.
  • Birds fly if and only if sky is clear.

 

Important Notes-

 

Note-01:

 

  • Negation ≡ NOT Gate of digital electronics.
  • Conjunction ≡ AND Gate of digital electronics.
  • Disjunction ≡ OR Gate of digital electronics.
  • Biconditional = EX-NOR Gate of digital electronics.

 

Note-02:

 

  • Each logical connective has some priority.
  • This priority order is important while solving questions.
  • The decreasing order of priority is-

 

 

Note-03:

 

  • Negation, Conjunction, Disjunction and Biconditional are both commutative and associative.
  • Conditional is neither commutative nor associative.

 

To gain better understanding about Logical Connectives,

Watch this Video Lecture

 

Next Article- Converting English Sentences To Propositional Logic

 

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