Cardinality/Examples/x^2-x
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Example of Cardinality
Let:
- $S_3 = \set {x^2 - x: x \in S_1}$
where $S_1 = \set {-1, 0, 1}$.
The cardinality of $S_3$ is given by:
- $\card {S_3} = 2$
Proof
The elements of $S_3$ can be found by calculation:
| \(\ds \paren {-1}^2 - \paren {-1}\) | \(=\) | \(\ds 1 + 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 2\) | ||||||||||||
| \(\ds 0^2 - 0\) | \(=\) | \(\ds 0 + 0\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) | ||||||||||||
| \(\ds 1^2 - 1\) | \(=\) | \(\ds 1 + 1\) | ||||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) |
Thus by definition of set:
- $S_3 = \set {0, 2}$
Thus $S_3$ has $2$ elements: $0, 2$.
Hence the result by definition of cardinality.
$\blacksquare$
Sources
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): Chapter $1$: Sets and Logic: Exercise $4$