Definition:Disjunction
Contents |
Definition
Disjunction is a binary connective written symbolically as $p \lor q$ whose behaviour is as follows:
- $p \lor q$
is defined as:
- Either $p$ is true or $q$ is true or possibly both.
This is called the disjunction (or logical alternation) of $p$ and $q$.
The statements $p$ and $q$ are known as the disjuncts.
$p \lor q$ is voiced:
- $p$ or $q$
Boolean Interpretation
From the above, we see that the boolean interpretations for $\mathbf A \lor \mathbf B$ under the model $\mathcal M$ are:
- $\left({\mathbf A \lor \mathbf B}\right)_{\mathcal M} = \begin{cases} T & : \mathbf A_{\mathcal M} = T \text{ or } \mathbf B_{\mathcal M} = T \\ F & : \text {otherwise} \end{cases}$
Complement
The complement of $\lor$ is the NOR operator.
Truth Function
The disjunction connective defines the truth function $f^\lor$ as follows:
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle f^\lor \left({F, F}\right)\) | \(=\) | \(\displaystyle F\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle f^\lor \left({F, T}\right)\) | \(=\) | \(\displaystyle T\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle f^\lor \left({T, F}\right)\) | \(=\) | \(\displaystyle T\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | |||
| \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle f^\lor \left({T, T}\right)\) | \(=\) | \(\displaystyle T\) | \(\displaystyle \) | \(\displaystyle \) | \(\displaystyle \) |
Truth Table
The truth table of $p \lor q$ and its complement is as follows:
$\begin{array}{|cc||c|c|} \hline p & q & p \lor q & p \downarrow q \\ \hline F & F & F & T \\ F & T & T & F \\ T & F & T & F \\ T & T & T & F \\ \hline \end{array}$
Notational Variants
Various symbols are encountered that denote the concept of disjunction:
| Symbol | Origin | Known as |
|---|---|---|
| $p \lor q$ | Bertrand Russell and Alfred North Whitehead: Principia Mathematica (1910) | vee or vel |
| $p\ \mathsf{OR} \ q$ | ||
| $p + q$ | ||
| $\operatorname A p q$ | Łukasiewicz's Polish notation |
Note
This usage of or, that allows the case where both disjuncts are true, is called inclusive or, or the inclusive disjunction.
Compare exclusive or.
Sources
- Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 2$: The Axiom of Specification
- Donald Kalish and Richard Montague: Logic: Techniques of Formal Reasoning (1964): $\text{II}: \S 1$
- E.J. Lemmon: Beginning Logic (1965): $\S 1.3$
- Alan G. Hamilton: Logic for Mathematicians (1978): $\S 1.2$
- D.J. O'Connor and Betty Powell: Elementary Logic (1980): $\S 1.12$ and Appendix
- Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.1$
- H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): $\S 1.1$
- Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems (2000): $\S 1.1, \ \S 1.4$ Fig. $1.9$
