Exercises
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Exercises
Let $R$ be the relation from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $x R y$ if and only if $3 | (x+y)$.
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Let $R$ be the relation from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $x R y$ if and only if $10 | (x-y)$.
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Let $R$ be the relation from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $x R y$ if and only if $x$ and $y$ have a common factor.
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Let $R$ be the relation from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $x R y$ if and only if both $x$ and $y$ are divisible by 3.
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Let $R$ be the relation from $\mathbb{R}$ to $\mathbb{R}$ defined by $x R y$ if and only if $x^2 + y^2 = 1$.
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Let $R$ be the relation from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $x R y$ if and only if $x + y$ is odd.
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Let $R$ be the relation from $\mathbb{R}$ to $\mathbb{R}$ defined by $x R y$ if and only if both $x - y$ is an integer.
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Explorations
Suppose $R$ and $S$ are relations both defined from $A$ to $B$. Define $R\cup S$ to be $\{(x, y)\in A\times B| xRy \text{ or } xSy\}$. In other words, it's the set of all ordered pairs that are in either of the two relations.
- Define $R$ and $S$ both from $\mathbb{Z^+}$ to $\mathbb{Z^+}$ by $xRy$ if and only of $x$ and $y$ are both even and $xSy$ if and only if $x$ and $y$ are both odd. Describe the ordered pairs in $R \cup S$.
- Define $R$ and $S$ both from $\mathbb{Z}$ to $\mathbb{Z}$ by $xRy$ if and only of $x-y$ is even and $xSy$ if and only if $x+y$ is even. Describe the ordered pairs in $R \cup S$.
- Show that the two relations in the previous question are equal to each other.
- Define $R$ and $S$ both from the set of all strings made up of 0's and 1's with less than five characters by $xRy$ if and only of $x$ and $y$ have the same length and $xSy$ if and only if $x$ and $y$ both start with a 1. Describe the ordered pairs in $R \cup S$.
Suppose $R$ and $S$ are relations both defined from $A$ to $B$. Define $R\cap S$ to be $\{(x, y)\in A\times B| xRy \text{ and } xSy\}$. In other words, it's the set of all ordered pairs that the two relations have in common.
- Define $R$ and $S$ both from $\mathbb{Z^+}$ to $\mathbb{Z^+}$ by $xRy$ if and only of $x$ and $y$ are both even and $xSy$ if and only if $x$ and $y$ are both odd. Describe the ordered pairs in $R \cap S$.
- Define $R$ and $S$ both from $\mathbb{Z}$ to $\mathbb{Z}$ by $xRy$ if and only of $3 | x-y$ is even and $xSy$ if and only if $5 | (x-y)$. Describe the ordered pairs in $R \cap S$.
- Define $R$ and $S$ both from $\mathbb{R}$ to $\mathbb{R}$ by $xRy$ if and only of $x < y$ have the same length and $xSy$ if and only if $x \le y$. Describe the ordered pairs in $R \cap S$.
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Let $R$ be the relation from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $x R y$ if and only if $3 | (x+y)$. Is $5 R 3$? Is $10 R (-1)$?
5 + 3 = 8 which is not divisible by three so (5, 3) is not part of the relation.
10 + (-1) = 9 which is divisible by three so (10, -1) is part of the relation and $10R(-1)$.
Let $R$ be the relation from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $x R y$ if and only if $3 | (x+y)$. Give all the numbers, $x$, such that $x R 5$.
We're looking for all the numbers, $x$, such that $3|(x + 5)$. Those are the numbers where
$$x + 5 = 3n, \forall n\in\mathbb{Z}$$So we want every number of the form $3n - 5$ or
$$\{x\in\mathbb{Z} | x =3n - 5, \forall n\in\mathbb{Z}\} = \{..., -5, -2, 1, 4, ...\}$$Let $R$ be the relation from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $x R y$ if and only if $10 | (x-y)$. Give 4 numbers, $x$, such that $x R 5$.
Any number that's 5 more than a multiple of 10 will work: $\{. . ., -15, -5, 5, 15, . . .\}$.
Let $R$ be the relation from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $x R y$ if and only if $x$ and $y$ have a common factor. Is $4 R 6$? Is $-2 R -7$?
4 and 6 have a common factor so $4 R 6$.
-2 and -7 don't have a common factor so $(-2, -7)$ isn't a part of the relation.
Let $R$ be the relation from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $x R y$ if and only if $x$ and $y$ have a common factor. Is it always true that $x R x$?
Every number has itself as a factor so it is true that $xRx$ for this relation.
Let $R$ be the relation from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $x R y$ if and only if both $x$ and $y$ are divisible by 3. Give all the numbers, $x$, such that $x R 5$.
Since 5 isn't divisible by 3, there are no numbers paired with 5 in the relation.
Let $R$ be the relation from $\mathbb{R}$ to $\mathbb{R}$ defined by $x R y$ if and only if $x^2 + y^2 = 1$. How many pairs of integers are in the relation?
The relation is all the pairs of points that lie on a unit circle. That graph has its maximum and minimum values at $\pm1$ so those are going to be the only integers in its domain along with 0 (the only integer between them). The corresponding $y$ values are also $\pm1$ so the only integer values are $(0, 1)$, $(0, -1)$, $(1, 0)$ and $(-1, 0)$.
Let $R$ be the relation from $\mathbb{R}$ to $\mathbb{R}$ defined by $x R y$ if and only if $x^2 + y^2 = 1$. Is it always true that $x R x$?
This isn't true, e.g. (0, 0) isn't in the relation. The only values for which this is true are the ones where
$$x^2 + x^2 = 1$$which only happes when $x=\sqrt{2}/2$ or $x=-\sqrt{2}/2$.
Let $R$ be the relation from $\mathbb{Z}$ to $\mathbb{Z}$ defined by $x R y$ if and only if $x + y$ is odd. Give all the numbers, $x$, such that $x R 6$
Since 6 is even $x$ can't be an even number since the sum of two even numbers is even. $x$ can be any odd number since the sum of an odd number and an even number is always odd.
Let $R$ be the relation from $\mathbb{R}$ to $\mathbb{R}$ defined by $x R y$ if and only if both $x - y$ is an integer. Is $4.1 R 5.2$? Is $3 R -4.1$?
4.1 - 5.2 = -1.1 which isn't an integer so $(4.1, 5.2)$ isn't in the relation.
3 - (-4.1) = 7.1 which also isn't an integer so $(3, -4.1)$ also isn't in the relation.
Let $R$ be the relation from $\mathbb{R}$ to $\mathbb{R}$ defined by $x R y$ if and only if both $x - y$ is an integer. Is it always true that if $x R y$ and $y R z$ then $x R z$?
Since $xRy$ is in the relation, $m = x - y$ must be an integer.
Since $yRz$ is in the relation, $n = y - z$ must be an integer.
Based on those result we have
$$x - z = (x - y) - (z - y) = m - n \in \mathbb{Z}$$Since thd difference of $x$ and $z$ is an integer, $xRz$.
Define $R$ and $S$ both from $\mathbb{Z}$ to $\mathbb{Z}$ by $xRy$ if and only of $x-y$ is even and $xSy$ if and only if $x+y$ is even. Describe the ordered pairs in $R \cup S$.
$R$ is the set of all pairs of integers whose difference is even where $S$ is the set of all pairs of integers whose sum is even. So the intersection is the set of all pairs of all numbers whose sum or difference is even.
Show that the two relations in the previous question are equal to each other.
We need to show is that $R = S$ but remember that these relations are actually just sets, specifically sets made up of ordered pairs so we can do this using a set-based proof.
Show that the two relations in the previous question are equal to each other.
We need to show is that $R = S$ but remember that these relations are actually just sets, specifically sets made up of ordered pairs so we can do this using a set-based proof.
First, let $(x, y) \in R$. By the definition of the relation $x - y$ is even. There are two possibilities here:
- $x$ and $y$ are both even. In this case, $-y$ must be even so $x -(-y)$ is also the difference of two even numbers and therefore even but $x - (-y) = x + y$ so $(x, y) \in S$.
- $x$ and $y$ are both odd. In this case, $-y$ must be odd so, by the same argument, $x + y$ is even and $(x, y) \in S$>
This shows that every $(x, y)\in R$ is also in $S$ so $R\subseteq S$.
Second, let $(x, y) \in S$ By the definition of the relation $x + y$ is even so, again, there are two possibilities. Either both numbers are even or both numbers are odd. In either case, $-y$ also has the same parity as $x$ so $x - y = x + (-y)$ is also even. This means that every $(x, y)\in S$ is also in $R$ so $S \subseteq R$ which completes the proof that $S = R$.
Define $R$ and $S$ both from $\mathbb{Z}$ to $\mathbb{Z}$ by $xRy$ if and only of $3 | x-y$ is even and $xSy$ if and only if $5 | (x-y)$. Describe the ordered pairs in $R \cap S$.
We need all the pairs of integers whose difference is divisible by 3 and whose difference is divisible by 5. The "numbers divisible by 5 and 3" is the same as "numbers divisible by 15" so that will be our intersection.
$$ R \cap S = \{(x, y) \in \mathbb{Z} \times \mathbb{Z} | 15 | (x - y)\}$$Define $R$ and $S$ both from $\mathbb{R}$ to $\mathbb{R}$ by $xRy$ if and only of $x < y$ have the same length and $xSy$ if and only if $x \le y$. Describe the ordered pairs in $R \cap S$.
We need all the ordered pairs of real numbers such that $x < y$ and $x \le y$. These are just all the numbers where $x < y$ since all of those pairs also satisfy $x \le y$ but the reverse isn't true. This means that
$$R \cap S = R$$