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What is the Definition of Set Image?

Published in Set Theory Function Image 3 mins read

The definition of the image of a set, often called the set image, describes how a function transforms a collection of elements from its domain.

Understanding the Image of a Set

Based on the provided definition, the image of a set is formally defined as follows:

Given a function f: A → B, which maps elements from set A (the domain) to set B (the codomain), and given a subset C ⊆ A (a part of the domain), the image of C under f is denoted by f(C).

The definition is:
f(C) = {f(x) ∣ x ∈ C}

In simpler terms, f(C) is the set containing all the results you get when you apply the function f to every single element that belongs to the subset C.

Key Points from the Definition

  • It applies to a subset (C) of the function's domain (A).
  • It collects the outputs (f(x)) for all inputs (x) that are in the specified subset C.
  • The result f(C) is a subset of the function's codomain (B).

Example

Let's consider a simple example to illustrate the concept:

  • Function f: f(x) = x²
  • Domain A: The set of all integers, ℤ
  • Codomain B: The set of all integers, ℤ
  • Subset C of the domain A: C = {-2, -1, 0, 1, 2}

To find the image of the set C under the function f, denoted as f(C), we apply the function f to each element in C:

  • f(-2) = (-2)² = 4
  • f(-1) = (-1)² = 1
  • f(0) = (0)² = 0
  • f(1) = (1)² = 1
  • f(2) = (2)² = 4

The set f(C) is the collection of these results. Note that we list each unique result only once in a set.

f(C) = {4, 1, 0, 1, 4}

Removing duplicates, we get:

f(C) = {0, 1, 4}

This set {0, 1, 4} is the image of the set C = {-2, -1, 0, 1, 2} under the function f(x) = x².

Visualizing the Set Image

Imagine the function f as an arrow mapping points from set A to set B. When you take a subset C within A, the set image f(C) consists of all the points in B that these elements in C point to.

Input (x ∈ C) Output (f(x) ∈ f(C))
-2 4
-1 1
0 0
1 1
2 4

The set image f(C) is simply the collection of all values in the 'Output' column, considered as a set.

This concept is fundamental in set theory and abstract algebra for understanding how functions behave with respect to subsets of their domains.

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