Logical connectives are used to combine statements and form new ones. They describe how the truth value of the compound statement depends on the individual statements. The most important logical connectives are negation, conjunction, disjunction, implication, and equivalence.

 

 

Negation

The negation of a statement expresses that the opposite is the case. If \( \large p \) is a statement, the negation is written as \( \large \lnot p \) and read "not p".

 

Examples:

 

• If \( \large p \) is "2 is an even number", then \( \large \lnot p \) is the statement "2 is not an even number".
• If \( \large p \) is "The sun is shining", then \( \large \lnot p \) is the statement "The sun is not shining".

 

$$ \begin{array}{|c|c|} \hline p & \lnot p \\ \hline T & F \\ F & T \\ \hline \end{array} $$

 

 

Conjunction (and)

The conjunction of two statements is true when both statements are true. If \( \large p \) and \( \large q \) are statements, the conjunction is written as \( \large p \land q \).

 

Examples:

 

• If \( \large p \) is "2 is an even number" and \( \large q \) is "2 is greater than 1", then \( \large p \land q \) is true.
• If \( \large p \) is "5 is a prime number" and \( \large q \) is "5 is an even number", then \( \large p \land q \) is false, because only the first statement is true.

 

$$ \begin{array}{|c|c|c|} \hline p & q & p \land q \\ \hline T & T & T \\ T & F & F \\ F & T & F \\ F & F & F \\ \hline \end{array} $$

 

 

Disjunction (or)

The disjunction of two statements is true when at least one of the statements is true. If \( \large p \) and \( \large q \) are statements, the disjunction is written as \( \large p \lor q \).

 

Note: In mathematics, "or" almost always means inclusive or, that is, the statement is also true if both are true.

 

Examples:

 

• If \( \large p \) is "2 is an even number" and \( \large q \) is "2 is greater than 10", then \( \large p \lor q \) is true, because at least one statement is true.
• If \( \large p \) is "5 is a prime number" and \( \large q \) is "5 is an even number", then \( \large p \lor q \) is true, because \( \large p \) is true.
• Only when both statements are false is the disjunction false.

 

$$ \begin{array}{|c|c|c|} \hline p & q & p \lor q \\ \hline T & T & T \\ T & F & T \\ F & T & T \\ F & F & F \\ \hline \end{array} $$

 

In everyday language, "or" is often used as exclusive or (either one or the other, but not both). In mathematics this can be written as a special operation: \( \large p \oplus q \). This is true exactly when one statement is true, but not both.

 

 

Implication (if … then …)

The implication expresses a conditional relationship. If \( \large p \) and \( \large q \) are statements, the implication is written as \( \large p \Rightarrow q \) and read "if p, then q".

 

The implication is only false when the premise \( \large p \) is true and the conclusion \( \large q \) is false. In all other cases it is considered true.

 

Examples:

 

• If \( \large p \) is "a number is divisible by 4" and \( \large q \) is "the number is divisible by 2", then \( \large p \Rightarrow q \) is true, because all numbers divisible by 4 are also divisible by 2.
• If \( \large p \) is "7 is an even number" and \( \large q \) is "10 is an even number", then \( \large p \Rightarrow q \) is true, because the premise is false – regardless of the conclusion.

 

This may seem counterintuitive in everyday language, but in mathematics it is practical: an implication with a false premise is counted as true, because the claim "if … then …" is not disproved.

 

$$ \begin{array}{|c|c|c|} \hline p & q & p \Rightarrow q \\ \hline T & T & T \\ T & F & F \\ F & T & T \\ F & F & T \\ \hline \end{array} $$

 

 

Equivalence (if and only if)

Equivalence expresses that two statements always have the same truth value. If \( \large p \) and \( \large q \) are statements, the equivalence is written as \( \large p \Leftrightarrow q \).

 

Examples:

 

• "A number is even if and only if it can be written as \( \large 2 \cdot k \) for an integer \( \large k \)".
• "A triangle is equilateral if and only if all its sides are equal in length".

 

The equivalence is true when both statements are true or both are false, and false when they have different truth values.

 

$$ \begin{array}{|c|c|c|} \hline p & q & p \Leftrightarrow q \\ \hline T & T & T \\ T & F & F \\ F & T & F \\ F & F & T \\ \hline \end{array} $$

 

 

Summary

The most important logical connectives are:

 

• Negation: \( \lnot p \) — true if \( p \) is false
• Conjunction: \( p \land q \) — true only if both are true
• Disjunction: \( p \lor q \) — true if at least one is true
• Implication: \( p \Rightarrow q \) — false only when p is true and q is false
• Equivalence: \( p \Leftrightarrow q \) — true when p and q have the same truth value

 

$$ \begin{array}{|c|c|c|c|c|c|c|} \hline p & q & \lnot p & p \land q & p \lor q & p \Rightarrow q & p \Leftrightarrow q \\ \hline T & T & F & T & T & T & T \\ T & F & F & F & T & F & F \\ F & T & T & F & T & T & F \\ F & F & T & F & F & T & T \\ \hline \end{array} $$