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mathematical connective symbols (Definition)

Introduction

Mathematics relies on precise symbolic language to express relationships between statements. Central to this language are connective symbols, also called logical connectives. These symbols combine simpler statements into more complex ones whose truth values depend in a well-defined way on their components.

Logical connectives form the foundation of mathematical reasoning, formal proof, set theory, computer science, and many areas of physics and engineering. Every theorem, definition, and proof ultimately rests on combinations of these connectives.

A statement that is either true or false is called a proposition. We typically denote propositions by lowercase letters such as $p$, $q$, and $r$.

Basic Logical Connectives

There are five primary logical connectives used throughout mathematics.

Negation

The negation of a proposition reverses its truth value.

$\displaystyle \neg p $

This is read as

“not $p$”

Example:

If

$\displaystyle p:$   “The number is positive”

then

$\displaystyle \neg p:$   “The number is not positive”

Conjunction

The conjunction of two propositions is true only if both are true.

$\displaystyle p \land q $

This is read as

“$p$ and $q$”

Example:

   “The number is positive and even”

Disjunction

The disjunction represents logical “or.”

$\displaystyle p \lor q $

This is the inclusive or, meaning one or both may be true.

Example:

   “The number is negative or zero”

Implication

The implication represents logical consequence.

$\displaystyle p \rightarrow q $

This is read as

“if $p$, then $q$”

Here:

  • $p$ is called the antecedent
  • $q$ is called the consequent

Example:

   If a number is divisible by 4, then it is even

Biconditional

The biconditional expresses logical equivalence.

$\displaystyle p \leftrightarrow q $

This is read as

“$p$ if and only if $q$”

This means both statements have the same truth value.

Example:

   A number is even if and only if it is divisible by 2

Truth Tables

The meaning of connectives is formally defined using truth tables.

Negation

\begin{displaymath} \begin{array}{c\vert c} p & \neg p \ \hline F & T \ T & F \end{array}\end{displaymath}

Conjunction

\begin{displaymath} \begin{array}{c\vert c\vert c} p & q & p \land q \ \hline F & F & F \ F & T & F \ T & F & F \ T & T & T \end{array}\end{displaymath}

Disjunction

\begin{displaymath} \begin{array}{c\vert c\vert c} p & q & p \lor q \ \hline F & F & F \ F & T & T \ T & F & T \ T & T & T \end{array}\end{displaymath}

Implication

\begin{displaymath} \begin{array}{c\vert c\vert c} p & q & p \rightarrow q \ \... ...F & F & T \ F & T & T \ T & F & F \ T & T & T \end{array}\end{displaymath}

Note the important fact:

An implication is false only when the antecedent is true and the consequent is false.

Biconditional

\begin{displaymath} \begin{array}{c\vert c\vert c} p & q & p \leftrightarrow q \... ...F & F & T \ F & T & F \ T & F & F \ T & T & T \end{array}\end{displaymath}

Compound Statements

Logical connectives allow the construction of complex propositions.

Example:

$\displaystyle (p \lor q) \land \neg r $

This represents

“($p$ or $q$) and not $r$”

To evaluate the compound proposition, we construct intermediate columns for each logical operation.

\begin{displaymath} \begin{array}{c\vert c\vert c\vert c\vert c\vert c} p & q & ... ...T & T & F & T & T & T \ T & T & T & T & F & F \ \end{array}\end{displaymath}

Operator Precedence

Logical connectives follow a precedence order similar to arithmetic operations.

\begin{displaymath} \begin{array}{c\vert l} \text{Highest} & \neg \ & \land \\... ... & \rightarrow \ \text{Lowest} & \leftrightarrow \end{array}\end{displaymath}

Parentheses should always be used when ambiguity is possible.

Logical Equivalence

Two propositions are logically equivalent if they always have the same truth value.

We write:

$\displaystyle p \equiv q $

Example: Double Negation

$\displaystyle \neg (\neg p) \equiv p $

Important Logical Laws

These laws are fundamental to mathematical reasoning.

Commutative Laws

$\displaystyle p \land q \equiv q \land p $

$\displaystyle p \lor q \equiv q \lor p $

Associative Laws

$\displaystyle (p \land q) \land r \equiv p \land (q \land r) $

$\displaystyle (p \lor q) \lor r \equiv p \lor (q \lor r) $

Distributive Laws

$\displaystyle p \land (q \lor r) \equiv (p \land q) \lor (p \land r) $

$\displaystyle p \lor (q \land r) \equiv (p \lor q) \land (p \lor r) $

De Morgan's Laws

$\displaystyle \neg (p \land q) \equiv \neg p \lor \neg q $

$\displaystyle \neg (p \lor q) \equiv \neg p \land \neg q $

These laws are essential for simplifying logical expressions.

Logical Connectives in Mathematics

Logical connectives appear in nearly every mathematical definition.

Example: Definition of Continuity

A function $f$ is continuous at $x_0$ if

$\displaystyle \forall \epsilon > 0, \exists \delta > 0 $

such that

$\displaystyle \vert x - x_0\vert < \delta \rightarrow \vert f(x) - f(x_0)\vert < \epsilon $

This definition uses implication as its central logical structure.

Applications

Logical connectives are fundamental in:

  • Mathematical proof
  • Computer programming
  • Digital circuit design
  • Set theory
  • Artificial intelligence

Every computer processor physically implements logical connectives using electronic logic gates.

Summary

Logical connectives provide the symbolic framework for mathematical reasoning.

The primary connectives are:

$\displaystyle \neg, \quad \land, \quad \lor, \quad \rightarrow, \quad \leftrightarrow $

They allow simple statements to be combined into complex logical structures and form the foundation of formal mathematics.

In more advanced mathematics these connectives will be extended through quantifiers, predicate logic, and formal proof systems.

Bibliography

1
Kenneth H. Rosen, Discrete Mathematics and Its Applications, 7th edition, McGraw Hill, 2012.
2
Susanna S. Epp, Discrete Mathematics with Applications, 4th Edition, Cengage Learning, 2011.
3
Patrick Suppes, Introduction to Logic, Dover Publications, 1999.
4
Elliott Mendelson, Introduction to Mathematical Logic, 5th Edition, Chapman and Hall/CRC, 2009.
5
Herbert B. Enderton, A Mathematical Introduction to Logic, 2nd Edition, Academic Press, 2001.
6
Paul R. Halmos, Naive Set Theory, Springer, 1974.
7
George Boolos, John Burgess, Richard Jeffrey, Computability and Logic, 5th Edition, Cambridge University Press, 2007.
8
Claude E. Shannon, “A Symbolic Analysis of Relay and Switching Circuits,” Transactions of the American Institute of Electrical Engineers, Vol. 57, 1938.



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Also defines:  negation, conjunction, disjunction, implication, antecedent, consequent, biconditional, truth tables, double negation

Cross-references: systems, programming, function, operation, proposition, theorem, computer

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