Cardinality of Set of All Mappings/Examples/2 Elements to 2 Elements
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Example of Cardinality of Set of All Mappings
Let $X = \set {a, b}$.
Let $Y = \set {u, v}$.
Then the mappings from $X$ to $Y$ are:
| \(\text {(1)}: \quad\) | \(\ds \map {f_1} a\) | \(=\) | \(\ds u\) | |||||||||||
| \(\ds \map {f_1} b\) | \(=\) | \(\ds v\) |
| \(\text {(2)}: \quad\) | \(\ds \map {f_2} a\) | \(=\) | \(\ds u\) | |||||||||||
| \(\ds \map {f_2} b\) | \(=\) | \(\ds u\) |
| \(\text {(3)}: \quad\) | \(\ds \map {f_3} a\) | \(=\) | \(\ds v\) | |||||||||||
| \(\ds \map {f_3} b\) | \(=\) | \(\ds v\) |
| \(\text {(4)}: \quad\) | \(\ds \map {f_4} a\) | \(=\) | \(\ds v\) | |||||||||||
| \(\ds \map {f_4} b\) | \(=\) | \(\ds u\) |
$f_1$ and $f_4$ are bijections.
$f_2$ and $f_3$ are neither surjections nor injections.
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $2$: Maps and relations on sets: Example $2.6$