# Definition:Exclusive Or

## Definition

**Exclusive Or** is a binary connective which can be written symbolically as $p \oplus q$ whose behaviour is as follows:

- $p \oplus q$

means:

**Either $p$ is true***or*$q$ is true*but not both*.

or symbolically:

- $p \oplus q := \paren {p \lor q} \land \neg \paren {p \land q}$

where $\land$ denotes the *and* operator and $\lor$ denotes the *or* operator.

There is no standard symbol for this, but the one shown above is seen commonly enough to be adopted as standard for this site.

### Truth Function

The exclusive or connective defines the truth function $f^\oplus$ as follows:

\(\ds f^\oplus \left({F, F}\right)\) | \(=\) | \(\ds F\) | ||||||||||||

\(\ds f^\oplus \left({F, T}\right)\) | \(=\) | \(\ds T\) | ||||||||||||

\(\ds f^\oplus \left({T, F}\right)\) | \(=\) | \(\ds T\) | ||||||||||||

\(\ds f^\oplus \left({T, T}\right)\) | \(=\) | \(\ds F\) |

### Truth Table

The characteristic truth table of the exclusive or operator $p \oplus q$ is as follows:

- $\begin{array}{|cc||c|} \hline p & q & p \oplus q \\ \hline \F & \F & \F \\ \F & \T & \T \\ \T & \F & \T \\ \T & \T & \F \\ \hline \end{array}$

### Boolean Interpretation

The truth value of $\mathbf A \oplus \mathbf B$ under a boolean interpretation $v$ is given by:

- $v \left({\mathbf A \oplus \mathbf B}\right) = \begin{cases} F & : v \left({\mathbf A}\right) = v \left({\mathbf B}\right) \\ T & : \text{otherwise} \end{cases}$

## Notational Variants

Various symbols are encountered that denote the concept of exclusive or:

Symbol | Origin | Known as |
---|---|---|

$p \oplus q$ | sometimes called o-plus
| |

$p\ \mathsf{XOR} \ q$ | ||

$p + q$ | ||

$p \not \Leftrightarrow q$ | ||

$p \not \equiv q$ | ||

$p \ne q$ | ||

$p \ \dot \lor \ q$ | ||

$p \ \_ \lor \ q$ |

## Also known as

This usage of **or**, that disallows the case where both disjuncts are true, is also called:

**exclusive disjunction****logical inequality****non-equivalence****symmetric difference**- the
**alternative function** **aut**(from the Latin), pronounced**out**.

Some sources refer to this as the **strong or**, where the **weak or** is used in the sense of the inclusive or.

In natural language, when it is necessary to be precise about the nature of the term being used, the phrase **but not both** is often employed.

## Examples

### Monday or Not Monday

The following is an example of an exclusive or statement:

*Either today is Monday or today is not Monday.*

### One of Five Statements is True

Which of the following $5$ statements is true?

- $(1): \quad$ Exactly one of these statements is false.

- $(2): \quad$ Exactly two of these statements are false.

- $(3): \quad$ Exactly three of these statements are false.

- $(4): \quad$ Exactly four of these statements are false.

- $(5): \quad$ Exactly five of these statements are false.

## Also see

- Results about
**exclusive or**can be found here.

## Sources

- 1946: Alfred Tarski:
*Introduction to Logic and to the Methodology of Deductive Sciences*(2nd ed.) ... (previous) ... (next): $\S \text{II}.7$: Sentential Calculus - 1959: A.H. Basson and D.J. O'Connor:
*Introduction to Symbolic Logic*(3rd ed.) ... (previous) ... (next): $\S 2.5$: Further Logical Constants - 1973: Irving M. Copi:
*Symbolic Logic*(4th ed.) ... (previous) ... (next): $2$ Arguments Containing Compound Statements: $2.1$: Simple and Compound Statements - 1980: D.J. O'Connor and Betty Powell:
*Elementary Logic*... (previous) ... (next): $\S \text{I}: 12$: Material Equivalence and Alternation - 1988: Alan G. Hamilton:
*Logic for Mathematicians*(2nd ed.) ... (previous) ... (next): $\S 1$: Informal statement calculus: $\S 1.2$: Truth functions and truth tables: Disjunction: Remark - 1993: M. Ben-Ari:
*Mathematical Logic for Computer Science*... (previous) ... (next): Chapter $2$: Propositional Calculus: $\S 2.1$: Boolean operators - 1993: Keith Devlin:
*The Joy of Sets: Fundamentals of Contemporary Set Theory*(2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.1$: What is a Set?: Exercise $1.1.1$ - 2000: Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*... (previous) ... (next): $\S 1.1$: Declarative sentences (footnote) - 2000: James R. Munkres:
*Topology*(2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 1$: Fundamental Concepts - 2008: David Joyner:
*Adventures in Group Theory*(2nd ed.) ... (previous) ... (next): Chapter $1$: Elementary, my dear Watson: $\S 1.1$: You have a logical mind if... - 2012: M. Ben-Ari:
*Mathematical Logic for Computer Science*(3rd ed.) ... (previous) ... (next): $\S 2.2.3$