2. For each of the following relations determine whether it is total, injective, surjective, bijective, a function? (a) $R: \mathbb{N} \to \mathbb{Z}$ $R(n) = \begin{cases} \frac{n}{2} & \text{if n is even} \\\\ \frac{1-n}{2} & \text{if n is odd} \end{cases}$ (b) $f: \mathbb{R} \to \mathbb{R}$ $R(x) = x \sin(x)$ (c) $g: \mathbb{R} \to \mathbb{R}$ $g(x) = y$ if and only if $x^2 = y$ (d) $h: \mathbb{N} \to \mathbb{N}$ $h(n) = \sqrt{n}$

          2. For each of the following relations determine whether it is total,
injective, surjective, bijective, a function?
(a) $R: \mathbb{N} \to \mathbb{Z}$ 		 $R(n) = \begin{cases} \frac{n}{2} & \text{if n is even} \\\\ \frac{1-n}{2} & \text{if n is odd} \end{cases}$
(b) $f: \mathbb{R} \to \mathbb{R}$ 		 $R(x) = x \sin(x)$
(c) $g: \mathbb{R} \to \mathbb{R}$ 		 $g(x) = y$ if and only if $x^2 = y$
(d) $h: \mathbb{N} \to \mathbb{N}$ 		 $h(n) = \sqrt{n}$

2. For each of the following relations determine whether it is total,
injective, surjective, bijective, a function?
(a) R: ℕ→ℤ R(n) = (n)/(2) if n is even

(1-n)/(2) if n is odd
(b) f: ℝ→ℝ R(x) = x sin(x)
(c) g: ℝ→ℝ g(x) = y if and only if x^2 = y
(d) h: ℕ→ℕ h(n) = √(n)

Added by David J.

Elementary and Intermediate Algebra

Alan S. Tussy, R. David Gustafson 5th Edition

Instant Answer

Solved by Expert Suzanne Harwood

Step 1

- Injective: No, since multiple natural numbers can map to the same integer (e.g. 1 and -1 both map to sin(π)). - Surjective: No, since not every integer can be reached by the relation (e.g. 1.5 cannot be reached). - Function: No, since there are multiple outputs Show more…

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2_ For each ofthe following relations determine whether it is total injective, surjective, bijective, a function? 1_ if n is even R(n) = if n is odd R(z) = x sin(x) g(z) = y if and only if &2 = y h(n) = Vn R: N _ Z (b) f : R= R (c) g : R= R (d) h :N _ N