# 0E: Exercises

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##### Exercise $$\PageIndex{1}$$

Define $$h : \mathbb{Z} \rightarrow \mathbb{Z}$$ by $$h(x) = x^2+4$$.  Determine (with reasons) whether or not $$h$$ is one−to−one and whether or not $$h$$ is onto.

##### Exercise $$\PageIndex{2}$$

Suppose $$f : A \rightarrow B$$ and $$g : B \rightarrow C$$ are functions.  Prove or disprove the following statements:

1. If $$g \circ f$$ is one-to-one then $$g$$ is one-to-one.

2. If $$g \circ f$$ is one-to-one and $$f$$ is onto, then $$g$$ is one-to-one.

3.  If $$g \circ f$$ is onto and  $$g$$is one-to-one, then  $$f$$ is onto.

##### Exercise $$\PageIndex{3}$$

Determine whether or not each of the following binary relations $$R$$ on the given set $$A$$ is reflexive, symmetric, antisymmetric, or transitive.  If a relation has a certain property, prove this is so; otherwise, provide a counterexample to show that it does not.  If $$R$$ is an equivalence relation, describe the equivalence classes of $$A$$.

1. Let $$S = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$$. Define a relation $$R$$ on $$A = S \times S$$ by $$(a, b) R (c, d)$$ if and only if $$10a + b \le 10c + d$$.

2. Let $$A = \mathbb{Z} \backslash \{0\}$$. Define a relation $$R$$ on $$A$$, by $$a R b$$ if and only if $$ab > 0$$.

3. Define a relation $$R$$ on $$A = \mathbb{Z}$$ by $$a R b$$ if and only if $$4 | (3a + b)$$.

4. Define a relation $$R$$ on $$A = \mathbb{Z}$$ by $$a R b$$ if and only if $$3 | (a^2 - b^2 )$$.

5. Let $$A = \mathbb{R}$$, If $$a,b \in \mathbb{R}$$, define $$a R b$$ if and only if $$a - b \in \mathbb{Z}$$.

6. Define a relation $$R$$ on the set $$\mathbb{Z} \times \mathbb{Z}$$ by $$(a, b) R (c, d)$$ if and only if $$ac = bd$$.

7. Define a relation $$R$$ on $$\mathbb{Z}$$ by $$a R b$$ if and only if $$2 \mid a^2+b$$. .

8. Define a relation $$R$$ on $$\mathbb{Z}$$ by $$a R b$$ if and only if $$5 | (2a + 3b)$$.

9. Let $$A = \mathbb{R}$$, If $$a,b \in \mathbb{R}$$, define $$a R b$$ if and only if $$a - b \in \mathbb{Q}$$.

This page titled 0E: Exercises is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Pamini Thangarajah.