Definition:Preimage

Relation

Let $\mathcal R \subseteq S \times T$ be a relation.

Let $\mathcal R^{-1} \subseteq T \times S$ be the inverse relation to $\mathcal R$, defined as:

$\mathcal R^{-1} = \left\{{\left({t, s}\right): \left({s, t}\right) \in \mathcal R}\right\}$

Preimage of an Element

Every $s \in S$ such that $\left({s, t}\right) \in \mathcal R$ is called a preimage of $t$.

In some contexts, it is not individual elements that are important, but all elements of $S$ which are of interest.

Thus the preimage (or inverse image) of an element $t \in T$ is defined as:

$\mathcal R^{-1} \left ({t}\right) := \left\{{s \in S: \left({s, t}\right) \in \mathcal R}\right\}$

This can also be written:

$\mathcal R^{-1} \left ({t}\right) := \left\{{s \in \operatorname{Im} \left({\mathcal R^{-1}}\right): \left({t, s}\right) \in \mathcal R^{-1}}\right\}$

That is, the preimage of $t$ under $\mathcal R$ is the image of $t$ under $\mathcal R^{-1}$.

The preimage of $t \in T$ is also known as the fiber of $t$.

Note that:

• $t \in T$ may have more than one preimage.
• It is possible for $t \in T$ to have no preimages at all, in which case $\mathcal R^{-1} \left ({t}\right) = \varnothing$.

Preimage of a Subset

Let $Y \subseteq T$.

The preimage (or inverse image) of $Y$ under $\mathcal R$ is defined as:

$\mathcal R^{-1} \left ({Y}\right) := \left\{{s \in S: \exists y \in Y: \left({s, y}\right) \in \mathcal R}\right\}$

That is, the preimage of $Y$ under $\mathcal R$ is the image of $Y$ under $\mathcal R^{-1}$.

Clearly:

$\displaystyle \mathcal R^{-1} \left ({Y}\right) = \bigcup_{y \in Y} \mathcal R^{-1} \left({y}\right)$

... the union of the preimages of each of the elements of $Y$.

If no element of $Y$ has a preimage, then $\mathcal R^{-1} \left ({Y}\right) = \varnothing$.

Preimage of a Relation

The preimage of a relation $\mathcal R \subseteq S \times T$ is:

$\operatorname{Im}^{-1} \left ({\mathcal R}\right) := \mathcal R^{-1} \left ({T}\right) = \left\{{s \in S: \exists t \in T: \left({s, t}\right) \in \mathcal R}\right\}$

Some sources, for example T.S. Blyth: Set Theory and Abstract Algebra (1975), call this the domain of $\mathcal R$.

Mapping

$\mathcal R$ can also be (and usually is in this context) a mapping or function.

Let $f: S \to T$ be a mapping.

Exactly the same notation and terminology concerning the concept of the preimage applies to its inverse $f^{-1}$.

Thus:

• Every $s \in S$ such that $f \left({s}\right) = t$ is called a preimage of $t$.

• The preimage (or inverse image) of an element $t \in T$ is defined as:

$f^{-1} \left ({t}\right) := \left\{{s \in S: f \left({s}\right) = t}\right\}$

This can also be written:

$f^{-1} \left ({t}\right) := \left\{{s \in \operatorname{Im} \left({f^{-1}}\right): \left({t, s}\right) \in f^{-1}}\right\}$

That is, the preimage of $t$ under $f$ is the image of $t$ under $f^{-1}$.

• The preimage of $Y \subseteq \operatorname{Im} \left({f}\right)$ is defined as:

$f^{-1} \left ({Y}\right) := \left\{{s \in S: \exists y \in Y: f \left({s}\right) = y}\right\}$

That is, the preimage of $Y$ under $f$ is the image of $Y$ under $f^{-1}$.

Clearly:

$\displaystyle f^{-1} \left ({Y}\right) = \bigcup_{y \in Y} f^{-1} \left({y}\right)$

... the union of the preimages of each of the elements of $Y$.

If no element of $Y$ has a preimage, then $\mathcal R^{-1} \left ({Y}\right) = \varnothing$.

The preimage of a mapping $f: S \to T$ is:

$\operatorname{Im}^{-1} \left ({f}\right) = f^{-1} \left ({T}\right) = \left\{{s \in S: \exists t \in T: f \left({s}\right) = t}\right\}$

Note that:

• From Preimages All Exist iff Surjection $f^{-1} \left({t}\right)$ is guaranteed not to be empty iff $f$ is a surjection.

• From Preimages All Unique iff Injection, if $f^{-1} \left({t}\right)$ is not empty, then it is guaranteed to be a singleton iff $f$ is an injection;

Thus, while $f^{-1}$ is always a relation, it is not actually a mapping unless $f$ is a bijection.

Alternative Notation

As well as using the notation $\operatorname{Im}^{-1} \left ({\mathcal R}\right)$ to denote the preimage of an entire relation, the symbol $\operatorname{Im}^{-1}$ can also be used as:

• For $t \in \operatorname{Im} \left({\mathcal R}\right)$, we have: $\operatorname{Im}^{-1}_\mathcal R \left ({t}\right) = \mathcal R^{-1} \left ({t}\right)$
• For $Y \subseteq \operatorname{Im} \left({\mathcal R}\right)$, we have: $\operatorname{Im}^{-1}_\mathcal R \left ({Y}\right) = \mathcal R^{-1} \left ({Y}\right)$

but this notation is clumsy and generally not preferred.

Also see

• Domain
• Codomain
• Range

• Image

Sources

• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 10$: Inverses and Composites
• Seth Warner: Modern Algebra (1965): $\S 12$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.4$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): $\text{I}$
• A.N. Kolmogorov and S.V. Fomin‎: Introductory Real Analysis (1968): $\S 1.3$
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 2.2$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 12$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 4, \ \S 5$
• W.A. Sutherland: Introduction to Metric and Topological Spaces (1975): Notation and Terminology
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.6$
• H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability (1996): Appendix $\text{A}.3$