# Mathematical Logic

Logic means reasoning. The reasoning may be a legal opinion or mathematical confirmation. We apply certain logic in Mathematics. Basic Mathematical logics are a negation, conjunction, and disjunction. The symbolic form of mathematical logic is, **‘~’ **for negation **‘^’** for conjunction and **‘ v ‘** for disjunction. In this article, we will discuss the basic Mathematical logic with the truth table and examples.

## Mathematical Logics Classification

Mathematical logic is classified into four subfields. They are:

- Set Theory
- Model Theory
- Recursion Theory
- Proof Theory

## Basic Mathematical Logical Operators

The three logical operators used in Mathematics are:

- Conjunction (AND)
- Disjunction (OR)
- Negation (NOT)

Let us discuss three types of logical operators in detail.

## Mathematical Logic Formulas

### Conjunction (AND)

We can join two statements by “**AND**” operand. It is also known as a conjunction. Its symbolic form is “**∧**“. In this operator, if anyone of the statement is false, then the result will be false. If both the statements are true, then the result will be true. It has two or more inputs but only one output.

**Truth Table for Conjunction (AND)**

Input |
Input |
Output |

A |
B |
A AND B (A ∧ B) |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

### Disjunction (OR)

We can join two statements by “OR” operand. It is also known as disjunction. It’s symbolic form is “∨”. In this operator, if anyone of the statement is true, then the result is true. If both the statements are false, then the result will be false. It has two or more inputs but only one output.

**Truth Table for Disjunction (OR)**

Input |
Input |
Output |

A |
B |
A OR B (A ∨ B) |

T | T | T |

T | F | T |

F | T | T |

F | F | F |

### Negation (NOT)

Negation is an operator which gives the opposite statement of the given statement. It is also known as **NOT**, denoted by **“∼”**. It is an operation that gives the opposite result. If the input is true, then the output will be false. If the input is false, then the output will be true. It has one input and one output. The truth table for NOT is given below:

Input |
Output |

A |
Negation A (∼A) |

T | F |

F | T |

### Mathematical Logics problems

**Example 1:**

Write the truth table values of conjunction for the given two statements

A: x is an even number

B: x is a prime number

**Solution:**

Given: A: x is an even number

B: x is a prime number

Let assume the different x values to prove the conjunction truth table

X value |
A |
B |
A AND B (A ∧ B) |

X = 2 | T | T | T |

X = 4 | T | F | F |

X = 3 | F | T | F |

X = 9 | F | F | F |

**Example 2: **

Write the truth table values of disjunction for the given two statements

A: p is divisible by 2

B: p is divisible by 3

**Solution:**

Given: A: P is divisible by 2

B: P is divisible by 3

Let assume the different x values to prove the disjunction truth table

Value of P |
A |
B |
A OR B (A ∨ B) |

P = 12 | T | T | T |

P = 4 | T | F | T |

P = 9 | F | T | T |

P = 7 | F | F | F |

**Example 3:**

Find the negation of the given statement “ a number 6 is an even number”

**Solution:**

Let “S” be the given statement

S = 6 is an even number

Therefore, the negation of the given statement is

∼S = 6 is not an even number.

Therefore, the negation of the statement is “ 6 is not an even number”

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