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Mathematical Logic

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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Scope and Subfields

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

(i) Set theory
(ii) Recursion theory
(iii) Model theory
(iv) Proof theory

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.

Basic Logical Operators

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There are three types of logic operators:

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

    NEGATION
    Negation is an operator which gives the opposite statement of the previous statement. It is also known as NOT, denoted by $\sim $. It is an operation that gives the opposite result. If input is true then output is false. If input is false then output is true. It has one input and one output. Truth table for NOT is given below :

    Input(P)Output(~P)

    Example of Negation

    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 “OR” operand. It is also known as Disjunction. It symbolic from is "$\vee $". In this operator if any one of the statement is true, then result is true, and if both the statements is false, then result will be false. It has two or more inputs and only one output.

    Truth Table for OR :

    InputInputOutput
    PQP OR Q
    TTT
    TFT
    FTT
    FFF


    Example of Disjunction

    Consider the following statements :
    A: 18+1 =19
    B: 180+1 = 1801
    and
    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. 

    ABA $\vee $ B
    TFT

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


    CONJUNCTION
    We can join two statements by “AND” operand. It is also known as Conjunction. Its symbolic from is "$\wedge $".In this operator If any one of the statement is false, then result will be false, and If both the statement are true, then only result will be true. It has two or more inputs and only one output.

    Truth table for AND (Conjunction):

    InputInputOutput
    PQP AND Q
    TTT
    TFF
    FTF
    FFF

    Example for Conjunction

    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.

    ABA $\wedge $ B
    TFF

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


    Examples

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

    Example 1 : What would be the negation of the statement "the number 8 is even"?

    Solution : Let the given statement is denoted by
    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.
    Example 2 : Two statements are given:
    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.

    Solution: The examples of truth values of conjunction are shown below.
    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.
    Example 3 : For the following statements, demonstrate all truth values of disjunction using examples -
    A: The number p is divisible by 2.
    B: The number p is divisible by 3.

    Solution: The examples of truth values of A OR B are listed below.
    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.

    Practice Problems

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    Problem 1: If S: The sum of x and y equals to a prime number. Write the negation of x.

    Problem 2: If P: x is an even number and Q: x is an odd number then write truth tables for P AND Q and P OR Q.
    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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