Definition:Set Difference

Definition

The (set) difference (or difference set) between two sets $S$ and $T$ is written $S \setminus T$, and means the set that consists of the elements of $S$ which are not elements of $T$:

$x \in S \setminus T \iff x \in S \land x \notin T$

It can also be defined as:

• $S \setminus T = \left\{{x \in S: x \notin T}\right\}$
• $S \setminus T = \left\{{x: x \in S \land x \notin T}\right\}$

$S \setminus T$ can be voiced:

• $S$ slash $T$
• $S$ cut down by $T$.

Another frequently seen notation for $S \setminus T$ is $S - T$. Both notations may be encountered on this website, but $\setminus$ is preferred.

Some authors refer to the expression $S \setminus T$ as the (relative) complement of $T$ in $S$, but the standard definition for the latter concept requires that $T \subseteq S$.

Illustration by Venn Diagram

The red area in the following Venn diagram illustrates $S \setminus T$:

Example

For example, if $S = \left\{{1, 2, 3}\right\}$ and $T = \left\{{2, 3, 4}\right\}$, then $S \setminus T = \left\{{1}\right\}$, while $T \setminus S = \left\{{4}\right\}$.

It can immediately be seen that $S \setminus T$ is not commutative, in general (and in fact, that it is anticommuntative).

See also

• Symmetric Difference

• Results about Set Difference can be found here.

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 5$: Complements and Powers
• W.E. Deskins: Abstract Algebra (1964): $\S 1.1$: Definition $1.2$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.6$
• Seth Warner: Modern Algebra (1965): $\S 3$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.2$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Introduction
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 1.2$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.2$: Ring Example $6$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 8$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 1$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 6$
• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 1.2$: Exercise $2$, $\S 1.4$: Footnote
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.2, \ \S 1.3$: Exercise $1.3.1 \ \text{(iii)}$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.2$