Chapter 01: Logic concepts

Operators

We are particularly interested in combining propositions by operators.

Definition

A compound proposition is a statement obtained by combining propositions with logical operators.

Conjunction

The conjunction operator is the binary operator which, when applied to two propositions and , yields the proposition “ and ”. This time, for to be true, we need both and to be true, note that is also commutative.

Truth Table of conjunction

T[1]

T

T

T

T

F

F

F

F[2]

T

F

F

F

F

F

F

Disjunction

The disjunction operator is the binary operator which, when applied to two propositions and , yields the proposition “ or ”. The disjunction operator returns T when at last one of the two propositions or is true. The operator is commutative[3].

Truth Table of disjunction

T

T

T

T

T

F

T

T

F

T

T

T

F

F

F

F

Note

More than two propositions can be joined using logical operators. In this instances, it is important to be careful about how they are grouped.

Example

  1. is true.

  2. is false.

Negation

The negation operator is a unary operator. Sometimes in mathematics it's important to determine what the opposite of a given mathematical statement is. This is usually referred to as "negating" a statement. One thing to keep in mind is that if a statement is true, then its negation is false (and if a statement is false, then its negation is true).

Truth Table of negation

T

F

F

T

Implication

In logic, implication is relationship between different propositions where the second proposition is a logical consequence of the first. The statement is denoted by . One way to think of the meaning of is to consider it a contract that says if the first condition is satisfied, then the second will also be satisfied. We say,

  1. implies .

  2. If then .

    Truth Table of implication

    T

    T

    F

    T

    T

    F

    F

    F

    F

    T

    T

    T

    F

    F

    T

    T

    the converse of is . The contapositive of is .

Equivalence

Logical equivalence is the condition of equality that exists between two statements or sentences in propositional logic. The relationship between the two statements translates verbally into "if and only if." In mathematics, logical equivalence is typically symbolized by a double arrow . The expression means .

Truth Table of equivalence

T

T

T

T

T

T

F

F

T

F

F

T

T

F

F

F

F

T

T

T

Equivalent propositions

Two propositions are equivalent if they have identical truth tables.

  1. T: True

  2. F: False

  3. Commutative

    property of an operation which allows you to change the order of the terms without changing the result.

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