Discrete Math
main function \(f :: D \rightarrow R\)
\(R\) is the CoDomain
1 Surjective
\[ {\color{blue}R \overset{inv}{\longrightarrow}} D \overset{f}{\longrightarrow} R \]
- Exists identity for Range (Right identity)
- by hitting our main function From behind
- \(f\) partitions it’s preimage \(D\) meaning \(inv\) must choose any element within it’s partition.
- Choice function
2 Injective
\[ D \overset{f}{\longrightarrow} R{\color{blue} \overset{inv}{\longrightarrow} D} \]
- Exists identity for Domain (Left identity)
- by Extending our main function with an inverse
3 Examples
3.1 Examples Surjective
\(Senators \overset{Elected}{\longrightarrow} PresidentialCandidates\)
\(PresidentialCandidates \overset{ElectedBy}{\longrightarrow} Senators\)
- In literary terms, passive voice inverts the active action.
- Senator Bob and Ruth Elected President Bush
- President Bush was Elected By Senator Bob and Ruth
3.2 Linear functions
All linear functions f(x) have f’’(x) = 0
4 Inverses
Given exponential function f(x), find g(x) st. f(g(x)) = kx with k as constant
We can solve using inflection point 2nd derivative method. We know that the second derivative of \[f(g(x))'' = 0\] \[(f'(g(x)) \cdot g'(x))' = 0\]
5 Even and Odd
\[\times \mapsto OR \] \[+ \mapsto XAND \]
XAND means true iff both parity are the same
\(\times \mapsto OR\)
| \(P\) | \(Q\) | \(P \times Q\) |
|---|---|---|
| E | E | E |
| E | O | E |
| O | E | E |
| O | O | O |
\(+ \mapsto XAND\)
| \(P\) | \(Q\) | \(P + Q\) |
|---|---|---|
| E | E | E |
| E | O | O |
| O | E | O |
| O | O | E |
- XAND is also known as XNOR
