In this page we are going to discuss about **mathematical logic **concept. Logic means reasoning. Reasoning may be legal opinion or mathematical confirmations. We apply certain logics in mathematics. Basic Mathematical logics are Negation, Conjunction and disjunction. Symbolic form of mathematical logics are, **‘~’ **for negation ** '`^^`'** for Conjunction and **' V '** for disjunction. We use this logics in physics there it is represented as AND, OR and NOT respectively. Truth table states the relation between the compound statement and sub – statement. In physics it is called as logic gates. A logic gate is simply an electronic circuit which operates on one or more input to produce an output.

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Basically, the mathematical logic has classified into following four subfields :

Although, each of these subfields share basic concepts of mathematical logic, yet they have different focuses. Mathematical logic was initially introduced for constructing fundamental frameworks for subjects like arithmetic, analysis and geometry. But it has now been utilized into several mathematical and non-mathematical areas. It is being outstandingly applied to computer science as well.

There are three types of logic operators:**Negation **(NOT)**Disjunction **(OR)**Conjunction **(AND)

Input(P) | Output(~P) |

T | F |

F | T |

The negation of the statement "This year is a leap year" is "This year is not a leap year.

Similarly, the negation of statement "Amar is taller than Vivek" may be written as either "Amar is not taller than Vivek" or "Amar is shorter than Vivek".

**DISJUNCTION**We can join two statements by

Input | Input | Output |

P | Q | P OR Q |

T | T | T |

T | F | T |

F | T | T |

F | F | F |

Consider the following statements :

A: 18+1 =19

B: 180+1 = 1801and

A: 10 + 1 = 11

B: 18 + 1 = 181

Here in both examples, the statements A are true and the statements B are false. Therefore A OR B is true.

A | B | A $\vee $ B |

T | F | T |

The disjunction of a compound statement is only false if both the combining statements are false else the disjunction is true.

CONJUNCTION

Input | Input | Output |

P | Q | P AND Q |

T | T | T |

T | F | F |

F | T | F |

F | F | F |

Consider above statements again:

A: 18+1 =19

B: 180+1 = 1801

and

A: 10 + 1 = 11

B: 18 + 1 = 181

Here, A AND B will be false, since statements A are true and statements B are false.

A | B | A $\wedge $ B |

T | F | F |

The conjunction of a compound statement is true only if both the combining statements are true else the conjunction would be false.

Have a look at the following examples of basic mathematical logics.

S = the number 8 is even.

The negation of S, i.e.

~S = the number 8 is **not** even.

According to the truth table of negation, if input is true, output must be false vice versa.

Here, S is true.

and ~S is false.

A: x is an even number.

B: x is a prime number.

List all the truth values of conjunction with the the help of examples.

If x = 2,

then statement A is true and statement B is true, therefore A AND B is also true.

If x = 4,

then statement A is true and statement B is false, therefore A AND B is false.

If x = 3,

then statement A is false and statement B is true, therefore A AND B is false.

If x = 9,

then statement A is false and statement B is false, therefore A AND B is also false.

A: The number p is divisible by 2.

B: The number p is divisible by 3.

If p = 12,

then statement A is true and statement B is true, therefore A OR B is true.

If x = 4,

then statement A is true and statement B is false, therefore A OR B is true.

If x = 9,

then statement A is false and statement B is true, therefore A OR B is true.

If x = 7,

then statement A is false and statement B is false, therefore A OR B is false.

More topics in Mathematical Logic | |

Negation | Conjunction |

Disjunction | Conditional Statement |

Compound Statement | Biconditional Statement |

Tautology | Truth Table |

Equivalence Statement | Directional Derivatives |

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