# 17.6: Exercises

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## Directed graph for a relation.

In each of Exercises 1–4, you are given a relation on a specific set. Draw a directed graph that represents the relation.

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

Relation $$\subsetneqq$$ on $$\mathscr{P}(\{a,b,c\})\text{.}$$

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

Relation $$\lt$$ on $$\{1,2,3,4\}\text{.}$$

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

Relation $$\equiv_3$$ on $$\mathbb{N}_{<13}\text{.}$$

##### Exercise $$\PageIndex{4}$$

Relation “has the same number of occurrences of the letter $$\mathrm{a}$$ as” on $$\Sigma^\ast_4$$ for alphabet $$\Sigma = \{a, z\}\text{.}$$

##### Exercise $$\PageIndex{5}$$

Recall that a relation on a set $$A$$ is just a subset of the Cartesian product $$A \times A\text{.}$$ Write out all relations on the set $$A = \{a,b\}$$ as subsets of $$A \times A\text{.}$$ Which of these relations are reflexive? Symmetric? Antisymmetric? Transitive?

## Testing reflexivity/symmetry/antisymmetry/transitivity.

In each of Exercises 6–17, you are given a relation on a specific set. Determine which of the properties reflexive, symmetric, antisymmetric, and transitive the given relation possesses.

##### Exercise $$\PageIndex{6}$$

Relation $$\lt$$ on $$\mathbb{R}\text{.}$$

##### Exercise $$\PageIndex{7}$$

Relation $$\mathord{\ge}$$ on $$\mathbb{R}\text{.}$$

##### Exercise $$\PageIndex{8}$$

Relation $$\mathord{\vert}$$ on $$\mathbb{Z}\text{.}$$

##### Exercise $$\PageIndex{9}$$

Relation $$\mathord{\subseteq}$$ on $$\mathscr{P}{X}\text{,}$$ where $$X$$ is an arbitrary, unspecified set.

##### Exercise $$\PageIndex{10}$$

Relation “is taller than” on the set of all living humans.

##### Exercise $$\PageIndex{11}$$

Relation “is parallel to” on the set of all straight lines in the plane.

##### Exercise $$\PageIndex{12}$$

Relation “is perpendicular to” on the set of all straight lines in the plane.

##### Exercise $$\PageIndex{13}$$

Relation “has the same length as” on $$\Sigma^\ast\text{,}$$ where $$\Sigma$$ is an arbitrary, unspecified alphabet set.

##### Exercise $$\PageIndex{14}$$

Relation “is shorter than” on $$\Sigma^\ast\text{,}$$ where $$\Sigma$$ is an arbitrary, unspecified alphabet set.

##### Exercise $$\PageIndex{15}$$

Relation “contains the same number of occurrences of the letter $$x$$ as” on $$\Sigma^\ast\text{,}$$ where $$\Sigma$$ is an arbitrary, unspecified alphabet set and $$x$$ is some fixed choice of letter in $$\Sigma\text{.}$$

##### Exercise $$\PageIndex{16}$$

Relation $$\Leftrightarrow$$ on the set of all logical statements involving the statement variables $$p_1,p_2,p_3,\ldots\text{.}$$

##### Exercise $$\PageIndex{17}$$

Relation $$R$$ defined by “$$a_1 \mathrel{R} a_2$$ if $$f(a_1) = f(a_2)$$” on a set $$A\text{,}$$ where $$f: A \rightarrow B$$ is an arbitrary, unspecified function.

## Properties of relations reflected in their graphs.

In each of Exercises 18–19, you are given a list of properties. Draw the directed graph of a relation on the set $$\{a,b,c,d\}$$ that possesses the given properties.

##### Exercise $$\PageIndex{18}$$

Symmetric and transitive, but neither reflexive nor antisymmetric.

##### Exercise $$\PageIndex{19}$$

Reflexive, antisymmetric, and transitive, but not symmetric.

##### Exercise $$\PageIndex{20}$$

Prove that a relation is symmetric if and only if it is equivalent to its own inverse relation.

##### Exercise $$\PageIndex{21}$$

As described in Section 17.3, the definition of antisymmetric relation can be formulated in symbolic language as

\begin{equation*} (\forall a_1 \in A)(\forall a_2 \in A)(a_1 \neq a_2 \Rightarrow a_1\ \not R\ a_2 \lor a_2\ \not R\ a_1) \text{.} \end{equation*}
Prove that each of the two conditionals provided in the Antisymmetric Relation Test are equivalent to the symbolic formulation of the definition of antisymmetric given above.

##### Exercise $$\PageIndex{22}$$

Suppose $$R$$ is a relation on a set $$A$$ that is both symmetric and antisymmetric. Prove that $$R$$ is a subset of the identity relation $$\{(x,x)\vert x \in A\}\text{.}$$

This page titled 17.6: Exercises is shared under a GNU Free Documentation License 1.3 license and was authored, remixed, and/or curated by Jeremy Sylvestre via source content that was edited to the style and standards of the LibreTexts platform.