Image of Subset under Mapping/Examples

From ProofWiki
Jump to: navigation, search

Examples of Images of Subsets under Mappings

Aribtrary Mapping from $\set {0, 1, 2, 3, 4, 5}$ to $\set {0, 1, 2, 3}$

Let:

\(\displaystyle S\) \(=\) \(\displaystyle \set {0, 1, 2, 3, 4, 5}\)
\(\displaystyle T\) \(=\) \(\displaystyle \set {0, 1, 2, 3}\)


Let $f: S \to S$ be the mapping defined as:

\(\displaystyle f \paren 0\) \(=\) \(\displaystyle 0\)
\(\displaystyle f \paren 1\) \(=\) \(\displaystyle 0\)
\(\displaystyle f \paren 2\) \(=\) \(\displaystyle 0\)
\(\displaystyle f \paren 3\) \(=\) \(\displaystyle 1\)
\(\displaystyle f \paren 4\) \(=\) \(\displaystyle 1\)
\(\displaystyle f \paren 5\) \(=\) \(\displaystyle 3\)


Let:

\(\displaystyle A\) \(=\) \(\displaystyle \set {0, 3}\)
\(\displaystyle B\) \(=\) \(\displaystyle \set {0, 1, 3}\)
\(\displaystyle C\) \(=\) \(\displaystyle \set {0, 1, 2}\)


Then:

\(\displaystyle f \sqbrk A\) \(=\) \(\displaystyle \set {0, 1}\)
\(\displaystyle f \sqbrk B\) \(=\) \(\displaystyle \set {0, 1}\)
\(\displaystyle f \sqbrk C\) \(=\) \(\displaystyle \set 0\)


and:

$\Img f = \set {0, 1, 3}$


Image of $\closedint {-3} 2$ under $x \mapsto x^4 - 1$

Let $f: \R \to \R$ be the mapping defined as:

$\forall x \in \R: f \paren x = x^4 - 1$

The image of the closed interval $\closedint {-3} 2$ is:

$f \sqbrk {\paren {\closedint {-3} 2} } = \closedint {-1} {80}$