- Negation
- Conjunction
- Disjunction
- Implications
- Reductions
- Equivalences

## → Truth value:-

The truth and falsity of a statement is called its truth value. The truth value of a statement is denoted by T or F according as it is true or false.

## → Negation:(not)

Let **p **be any preposition. The negative of given proposition **P **denoted by **⁓p** is called negation of p.

**Example: **

**p**-“I love animals.”**negation of p** i.e **⁓p** is “I do not love animals.”

p: The summer in Terai is very hot.

⁓p: The summer in Terai is not very hot.

## Truth Table

p | ⁓p |

T | F |

F | T |

**In this truth table:**

“

**p**” represents the truth value of proposition.

“

**⁓p**” represents the negation of

**p**.

## → Conjunction:(AND)

Given two propositions **p** and **q**, the proposition “p and q” denoted ny **pΛq** is the proposition that is true whenever both the propositions p and q are true, false otherwise.

• The proposition that is obtained by the use of “**and**” operator is also called conjunction of p and q.

**Example:**

If we have propositions **p**=”Ram is smart.” and **q**=”Ram is intelligent.”

then conjunction of p and q is **(pΛq):** Ram is smart and intelligent.

• This proposition is true only when Ram is smart and intelligent also, false otherwise.

## Truth Table

p | q | pΛq |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

**In this truth table:**

“

**p**and

**q**” represents the truth value of proposition.

“

**pΛq**” represents the conjunction (AND) operation between p and q.

## → Disjunction:(OR)

Given two propositions **p **and **q**, the proposition “**p OR q**” is denoted by **p⋁q** is the proposition that is false whenever both the propositions p and q are false, true otherwise.

• The proposition that is obtained by the use of “or” operator is also called disjunction of p and q.

**Example:**

if we have proposition **p**= Ram is intelligent and **q**= Ram is diligent.

then disjunction of p and q is** (p⋁q)**: Ram is intelligent or diligent.

## Truth Table

p | q | p⋁q |

T | T | T |

T | F | T |

F | T | T |

F | F | F |

**In this truth table:**

“

**p**and

**q**” represents the truth value of proposition.

“

**p**” represents the disjunction (OR) operation between p and q.

**⋁**q## → Exclusive OR:(XOR)

Given two propositions **p **and **q**, the proposition exclusive or of p and q is denoted by **p⨁q** is the proposition that is true whenever only onen of the propositions p and q is true, false otherwise.

**Example:**

if we have propositins **p**= Ram drinks coffee in the morning. and **q**= Ram drinks tea in the morning. then the exclusive or of p and q **(p⨁q)**: Ram drinks coffee or tea in the morning.

## Truth Table

p | q | p⨁q |

T | T | T |

T | F | T |

F | T | T |

F | F | F |

**In this truth table:**

“

**p**and

**q**” represents the truth value of proposition.

“

**p**” represents the exclusive or (XOR) operation between p and q.

**⨁**q## → Implication(**→**):

Given two propositions p and q, the implication p→q is the proposition that is false when p is true and q is false, true otherwise.

•Here **p **is called *hypothesis *and **q **is called *consequence*.

**We use different terminologies to express p→q like:**

• if **p **, then **q**

• **q **is the consequence of **p**

• **p **only if **q**

**Example**:

if we have propositions **p**=Today is sunday. and **q**=It is hot today. then **implication of p and q (p→q )**: If today is sunday then it is hot day.**or **Today is sunday only if it is hot today.

## Truth Table

p | q | p→q |

T | T | T |

T | F | F |

F | T | T |

F | F | T |

**In this truth table:**

• “

**p**” and “

**q**” represent the truth values of the hypothesis and conclusion.

• “

**p→q**” represents the truth value of the implication.