103 lines
3 KiB
Markdown
103 lines
3 KiB
Markdown
Page 515
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**Test Yourself**
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1. If $R$ is a relation from $A$ to $B$, $x \in A$, and $y \in B$, the notation
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$x R y$ means that ____.
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$x$ is related to $y$ by $R$
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2. If $R$ is a relation from $A$ to $B$, $x \in A$, and $y \in B$, the notation
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$x \cancel{R} y$ means that ____.
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$x$ is not related to $y$ by $R$.
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3. If $R$ is a relation from $A$ to $B$, $x \in A$, and $y \in B$, the notation
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$(y, x) \in R^{-1}$ if, and only if, ____.
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$$ (x, y) \in R $$
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4. A relation on a set $A$ is a relation from ____ to ____.
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$A$; $A$
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5. If $R$ is a relation on a set $A$, the directed graph of $R$ has an arrow
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from $x$ to $y$ if, and only if, ____.
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$x$ is related to $y$ by $R$
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---
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Page 526
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**Test Yourself**
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1. For a relation $R$ on a set $A$ to be reflexive means that ____.
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$\forall x \in A, x R x$
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2. For a relation $R$ on a set $A$ to be symmetric means that ____.
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$\forall x, y \in A, x R y \to y R x$
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3. For a relation $R$ on a set $A$ to be transitive means that ____.
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$\forall x, y, z \in A, (x R y \wedge y R z) \to x R z$
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4. To show that a relation $R$ on an infinite set $A$ is reflexive, you suppose
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that ____ and you show that ____.
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$x \in A$; $x R x$
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5. To show that a relation $R$ on an infinite set $A$ is symmetric, you suppose
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that ____ and you show that ____.
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$\forall x, y \in A, x R y$; $y R x$
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6. To show that a relation $R$ on an infinite set $A$ is transitive, you suppose
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that ____ and you show that ____.
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$\forall x, y, z \in A, x R y \wedge y R z$; $x R z$
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7. To show that a relation $R$ on a set $A$ is not reflexive, you ____.
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$\exists x \in A, x \cancel{R} x$
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8. To show that a relation $R$ on a set $A$ is not symmetric, you ____.
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$\exists x, y \in A, x R y \to y \cancel{R} x$
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9. To show that a relation $R$ on a set $A$ is not transitive, you ____.
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$\exists x, y, z \in A, (x R y \wedge y R z) \to x \cancel{R} z$
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10. Given a relation $R$ on a set $A$, the transitive closure of $R$ is the
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relation $R^t$ on $A$ that satisfies the following three properties: ____,
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____, and ____.
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$R^t$ is transitive; $R \subseteq R^t$; if $S$ is any other transitive relation
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that contains $R$, then $R^t \subseteq S$
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---
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Page 543
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**Test Yourself**
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1. For a relation on a set to be an equivalence relation, it must be ____.
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2. The notation $m \equiv n (\mod d)$ is read "____" and means that ____.
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3. Given an equivalence relation $R$ on a set $A$ and given an element $a$ in
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$A$, the equivalence class of $a$ is denoted ____ and is defined to be ____.
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4. If $A$ is a set, $R$ is an equivalence relation on $A$, and $a$ and $b$ are
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elements of $A$, then either $[a] = [b]$ or ____.
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5. If $A$ is a set and $R$ is an equivalence relation on $A$, then the distinct
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equivalence classes of $R$ form ____.
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6. Let $A = \mathbb{Z} \times (\mathbb{Z} - \{0\})$, and define a relation $R$
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on $A$ by specifying that for every $(a, b)$ and $(c, d)$ in $A$,
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$(a, b) R (c, d)$ if, and only if, $ad = bc$. Then there is exactly one
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equivalence class of $R$ for each ____.
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