Definition:Image (Relation Theory)/Relation/Relation/General Definition
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Definition
Let $\ds \prod_{i \mathop = 1}^n S_i$ be the cartesian product of sets $S_1$ to $S_n$.
Let $\ds \RR \subseteq \prod_{i \mathop = 1}^n S_i$ be an $n$-ary relation on $\ds \prod_{i \mathop = 1}^n S_i$.
The image of $\RR$ is the set defined as:
- $\Img \RR := \set {s_n \in S_n: \exists \tuple {s_1, s_2, \ldots, s_{n - 1} } \in \ds \prod_{i \mathop = 1}^{n - 1} S_i: \tuple {s_1, s_2, \ldots, s_n} \in \RR}$
The concept is usually encountered when $\RR$ is an endorelation on $S$:
- $\Img \RR := \set {s_n \in S: \exists \tuple {s_1, s_2, \ldots, s_{n - 1} } \in S^{n - 1}: \tuple {s_1, s_2, \ldots, s_n} \in \RR}$
Technical Note
The $\LaTeX$ code for \(\Img {f}\) is \Img {f}
.
When the argument is a single character, it is usual to omit the braces:
\Img f
Sources
- 1993: Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (2nd ed.) ... (previous) ... (next): $\S 1$: Naive Set Theory: $\S 1.6$: Functions