**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,

**Next Article-** **Converting English Sentences To Propositional Logic**

Get more notes and other study material of **Propositional Logic**.