# Truth Tables of Five Common Logical Connectives or Operators

In this lesson, we are going to construct the **truth tables** of the five (5) common logical connectives or operators. They are considered common logical connectives because they are very popular, useful and always taught together.

Before we begin, I suggest that you review my other lesson in which the link is shown below.

This introductory lesson about truth tables contains prerequisite knowledge or information that will help you better understand the content of this lesson.

Introduction to Truth Tables, Statements and Connectives

Le’s start by listing the five (5) common logical connectives.

## The Five (5) Common Logical Connectives or Operators

- Logical Negation
- Logical Conjunction (AND)
- Logical Disjunction (Inclusive OR)
- Logical Implication (Conditional)
- Logical Biconditional (Double Implication)

## I. Truth Table of Logical Negation

The **negation** of a statement is also a statement with a truth value that is exactly opposite that of the original statement. For instance, the negation of the statement is written symbolically as

~[latex]\large{P}[/latex] or [latex]\large{\neg P}[/latex].

~[latex]{P}[/latex] or [latex]{\neg P}[/latex] is read as “not [latex]P[/latex].”

**Remember:** The negation operator denoted by the symbol **~** or [latex]\neg[/latex] takes the truth value of the original statement then output the exact opposite of its truth value. In other words, negation simply reverses the truth value of a given statement. Thus, if statement [latex]P[/latex] is true then the truth value of its negation is false. In the same manner if [latex]P[/latex] is false the truth value of its negation is true.

## II. Truth Table of Logical Conjunction

A **conjunction** is a type of compound statement that is comprised of two propositions (also known as simple statements) joined by the AND operator.

The symbol that is used to represent the AND or logical conjunction operator is [latex]\color{red}\Large{\wedge}[/latex]. It looks like an inverted letter V.

If we have two simple statements [latex]P[/latex] and [latex]Q[/latex], and we want to form a compound statement joined by the AND operator, we can write it as:

[latex]\large{P \wedge Q}[/latex].

[latex]{P \wedge Q}[/latex] is read as “[latex]P[/latex] and [latex]Q[/latex].”

**Remember:** The truth value of the compound statement [latex]P \wedge Q[/latex] is only true if the truth values [latex]P[/latex] and [latex]Q[/latex] are both true. Otherwise, [latex]P \wedge Q[/latex] is false.

Notice in the truth table below that when [latex]P[/latex] is true and [latex]Q[/latex] is true, [latex]P \wedge Q[/latex] is true. However, the other three combinations of propositions [latex]P[/latex] and [latex]Q[/latex] are false.

## III. Truth Table of Logical Disjunction

A **disjunction** is a kind of compound statement that is composed of two simple statements formed by joining the statements with the OR operator.

In a disjunction statement, the use of OR is inclusive. That means “one or the other” or both.

The symbol that is used to represent the OR or logical disjunction operator is [latex]\color{red}\Large{ \vee }[/latex]. It resembles the letter V of the alphabet.

Two propositions [latex]P[/latex] and [latex]Q[/latex] joined by OR operator to form a compound statement is written as:

[latex]\large{P \vee Q}[/latex].

[latex]{P \vee Q}[/latex] is read as “[latex]P[/latex] or [latex]Q[/latex].”

**Remember:** The truth value of the compound statement [latex]P \vee Q[/latex] is true if the truth value of either the two simple statements [latex]P[/latex] and [latex]Q[/latex] is true. Moreso, [latex]P \vee Q[/latex] is also true when the truth values of both statements [latex]P[/latex] and [latex]Q[/latex] are true. However, the only time the disjunction statement [latex]P \vee Q[/latex] is false, happens when the truth values of both [latex]P[/latex] and [latex]Q[/latex] are false.

## IV. Truth Table of Logical Implication

An **implication** (also known as a **conditional statement**) is a type of compound statement that is formed by joining two simple statements with the logical implication connective or operator.

The symbol that is used to represent the logical implication operator is an arrow pointing to the right, thus a rightward arrow.

When two simple statements [latex]P[/latex] and [latex]Q[/latex] are joined by the implication operator, we have:

[latex]\Large{P \to Q}[/latex].

- where [latex]P[/latex] is known as the
**hypothesis**

- where [latex]Q[/latex] is known as the
**conclusion**

There are many ways how to read the conditional [latex]{P \to Q}[/latex]. Below are some of the few common ones.

[latex]{P \to Q}[/latex] is read as “[latex]P[/latex] implies [latex]Q[/latex]”.

[latex]{P \to Q}[/latex] is read as “If [latex]P[/latex] then [latex]Q[/latex]”.

[latex]{P \to Q}[/latex] is read as “[latex]P[/latex] only if [latex]Q[/latex]”.

[latex]{P \to Q}[/latex] is read as “If [latex]P[/latex] is sufficient for [latex]Q[/latex]”.

[latex]{P \to Q}[/latex] is read as “[latex]Q[/latex] is necessary for [latex]P[/latex]”.

[latex]{P \to Q}[/latex] is read as “[latex]Q[/latex] follows from [latex]P[/latex]”.

[latex]{P \to Q}[/latex] is read as “[latex]Q[/latex] if [latex]P[/latex]”.

**Remember:** The truth value of the compound statement [latex]P \to Q[/latex] is true when both the simple statements [latex]P[/latex] and [latex]Q[/latex] are true. Moreso, [latex]P \to Q[/latex] is always true if [latex]P[/latex] is false. The only scenario that [latex]P \to Q[/latex] is false happens when [latex]P[/latex] is true, and [latex]Q[/latex] is false.

## V. Truth Table of Logical Biconditional or Double Implication

A **double implication** (also known as a **biconditional statement**) is a type of compound statement that is formed by joining two simple statements with the biconditional operator. A biconditional statement is really a combination of a conditional statement and its converse.

The biconditional operator is denoted by a double-headed arrow.

When you join two simple statements (also known as molecular statements) with the biconditional operator, we get:

[latex]\Large{P \leftrightarrow Q}[/latex]

[latex]{P \leftrightarrow Q}[/latex] is read as “[latex]P[/latex] if and only if [latex]Q[/latex].”

- where [latex]P[/latex] is known as the antecedent

- where [latex]Q[/latex] is known as the consequent

**Remember:** The truth value of the biconditional statement [latex]P \leftrightarrow Q[/latex] is true when both simple statements [latex]P[/latex] and [latex]Q[/latex] are both true or both false. Otherwise, [latex]P \leftrightarrow Q[/latex] is false.

**You may also be interested in these related math lessons or tutorials:**

Truth Tables Practice Problems with Answers

Introduction to Truth Tables, Statements, and Logical Connectives

Converse, Inverse, and Contrapositive of a Conditional Statement