Gibbs Stanley
03/04/2024 · Primary School
e. Berikan masing-masing dua contoh pemetaan yang : \( \begin{array}{lll}\text { (a). surjektif } & \text { (b). injektif } & \text { (c). bijektif }\end{array} \) f. Misalkan \( \theta: A \rightarrow B \) dan \( \sigma: B \rightarrow C \quad \) masing-masing pemetaan, buktikan apakah \( \begin{array}{l}\text { (i). Jika } \theta \text { dan } \sigma \text { surjektif maka } \sigma 0 \theta \text { pemetaan surjektif }\end{array} \) (ii). Jika \( \theta \) dan \( \sigma \) injektif maka \( \sigma 0 \theta \) pemetaan injektif
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### E. Contoh Pemetaan
#### (a) Pemetaan Surjektif
1. \( f(x) = x^3 \)
2. \( g(x) = e^x \) (pada interval \( (0, \infty) \))
#### (b) Pemetaan Injektif
1. \( h(x) = 2x + 1 \)
2. \( k(n) = 3n \)
#### (c) Pemetaan Bijektif
1. \( m(x) = x + 5 \)
2. \( n(n) = n + 1 \)
### F. Pembuktian Pemetaan
#### (i) Jika \( \theta \) dan \( \sigma \) surjektif, maka \( \sigma \circ \theta \) adalah pemetaan surjektif.
**Pembuktian:**
- Karena \( \sigma \) surjektif, ada \( b \in B \) sehingga \( \sigma(b) = y \).
- Karena \( \theta \) surjektif, ada \( a \in A \) sehingga \( \theta(a) = b \).
- Dengan demikian, \( \sigma(\theta(a)) = y \), yang menunjukkan bahwa \( \sigma \circ \theta \) surjektif.
#### (ii) Jika \( \theta \) dan \( \sigma \) injektif, maka \( \sigma \circ \theta \) adalah pemetaan injektif.
**Pembuktian:**
- Misalkan \( \sigma \circ \theta(a_1) = \sigma \circ \theta(a_2) \).
- Maka, \( \sigma(\theta(a_1)) = \sigma(\theta(a_2)) \).
- Karena \( \sigma \) injektif, \( \theta(a_1) = \theta(a_2) \).
- Karena \( \theta \) injektif, \( a_1 = a_2 \).
- Dengan demikian, \( \sigma \circ \theta \) injektif.
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