Lesson test: Equivalence relations

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Equivalence relations

1.[3p]
Which of these relations on $ℤ$ are equivalence relations?

Select all that apply

$a \sim b$ when $a \equiv b (mod 7)$

Correct

$a \sim b$ when $a^{2} = b^{2}$

Correct

$a \sim b$ when $a + b$ is even

Correct

$a \sim b$ when $a ∣ b$

The answer is: $a \sim b$ when $a \equiv b \pmod{7}$, $a \sim b$ when $a^2 = b^2$, $a \sim b$ when $a + b$ is even

$a \sim b$ when $| a - b | \leq 1$

The answer is: $a \sim b$ when $a \equiv b \pmod{7}$, $a \sim b$ when $a^2 = b^2$, $a \sim b$ when $a + b$ is even

$a \sim b$ when $a b > 0$

The answer is: $a \sim b$ when $a \equiv b \pmod{7}$, $a \sim b$ when $a^2 = b^2$, $a \sim b$ when $a + b$ is even

2.[2p]
Match each relation on $ℝ$ to the property it lacks.

$x \leq y$
$| x - y | \leq 1$
$x y > 0$
$x - y \in ℤ$

reflexive
symmetric
transitive
none, it is an equivalence relation

Show the answer

$x \leq y$ : symmetric $| x - y | \leq 1$ : transitive $x y > 0$ : reflexive $x - y \in ℤ$ : none, it is an equivalence relation

3.[2p]
What is wrong with the argument "if $a \sim b$ then $b \sim a$ by symmetry, so $a \sim a$ by transitivity, hence every symmetric and transitive relation is reflexive"?

It assumes some $b$ with $a \sim b$ exists, and for some $a$ there may be none

Correct

Transitivity cannot be applied to $a \sim b$ and $b \sim a$ together

The answer is: It assumes some $b$ with $a \sim b$ exists, and for some $a$ there may be none

Symmetry only applies when $a \neq b$

The answer is: It assumes some $b$ with $a \sim b$ exists, and for some $a$ there may be none

Nothing: every symmetric and transitive relation is reflexive

The answer is: It assumes some $b$ with $a \sim b$ exists, and for some $a$ there may be none

4.[1p]
How many relations are there on a set with $3$ elements?

CorrectNot quite: 512

5.[2p]
How many equivalence relations are there on the set ${1, 2, 3, 4}$ ?

CorrectNot quite: 15

6.[1p]
Which set is the congruence class $[2]$ modulo $3$ ?

${\dots, - 4, - 1, 2, 5, 8, \dots}$

Correct

${\dots, - 2, 2, 6, 10, \dots}$

The answer is: $\{ \ldots, -4, -1, 2, 5, 8, \ldots \}$

${\dots, - 5, - 2, 1, 4, 7, \dots}$

The answer is: $\{ \ldots, -4, -1, 2, 5, 8, \ldots \}$

${\dots, - 6, - 3, 0, 3, 6, \dots}$

The answer is: $\{ \ldots, -4, -1, 2, 5, 8, \ldots \}$

7.[2p]
Put the lines of the proof that the relation $(a, b) \sim (c, d)$ when $a d = b c$ , on pairs with nonzero second coordinate, is transitive in order.

Let $(a, b) \sim (c, d)$ and $(c, d) \sim (e, f)$ , so $a d = b c$ and $c f = d e$ .
Multiplying the first equation by $f$ gives $a d f = b c f$ .
Substituting $c f = d e$ gives $a d f = b d e$ .
Since $d \neq 0$ , $a f = b e$ , that is, $(a, b) \sim (e, f)$ .
So $d (a f - b e) = 0$ .

Show the answer

a, b, c, d, e

8.[2p]
In building $ℚ$ from pairs $(a, b)$ with $(a, b) \sim (c, d)$ when $a d = b c$ , why must the pair $(0, 0)$ be excluded?

It would be related to every pair, and transitivity would fail

Correct

It would make the relation fail to be reflexive

The answer is: It would be related to every pair, and transitivity would fail

It would make the relation fail to be symmetric

The answer is: It would be related to every pair, and transitivity would fail

It would give $ℚ$ a second zero, but the relation would still be an equivalence

The answer is: It would be related to every pair, and transitivity would fail

9.[3p]
Which of these rules are well defined?

Select all that apply

$[a] + [b] = [a + b]$ on congruence classes modulo $n$

Correct

$[(a, b)] \mapsto [(a^{2}, b^{2})]$ on $ℚ$

Correct

The congruence class of $a$ modulo $6$ sent to the class of $a$ modulo $3$

Correct

$[(a, b)] \mapsto a$ from $ℚ$ to $ℤ$

The answer is: $[a] + [b] = [a + b]$ on congruence classes modulo $n$, $[(a, b)] \mapsto [(a^2, b^2)]$ on $ℚ$, The congruence class of $a$ modulo $6$ sent to the class of $a$ modulo $3$

$[(a, b)] \oplus [(c, d)] = [(a + c, b + d)]$ on $ℚ$

The answer is: $[a] + [b] = [a + b]$ on congruence classes modulo $n$, $[(a, b)] \mapsto [(a^2, b^2)]$ on $ℚ$, The congruence class of $a$ modulo $6$ sent to the class of $a$ modulo $3$

The congruence class of $a$ modulo $3$ sent to the class of $a$ modulo $6$

The answer is: $[a] + [b] = [a + b]$ on congruence classes modulo $n$, $[(a, b)] \mapsto [(a^2, b^2)]$ on $ℚ$, The congruence class of $a$ modulo $6$ sent to the class of $a$ modulo $3$

1.[3p] Which of these relations on ℤ are equivalence relations?

2.[2p] Match each relation on ℝ to the property it lacks.

3.[2p] What is wrong with the argument "if a∼b then b∼a by symmetry, so a∼a by transitivity, hence every symmetric and transitive relation is reflexive"?

4.[1p] How many relations are there on a set with 3 elements?

5.[2p] How many equivalence relations are there on the set {1,2,3,4}?

6.[1p] Which set is the congruence class [2] modulo 3?

7.[2p] Put the lines of the proof that the relation (a,b)∼(c,d) when ad=bc, on pairs with nonzero second coordinate, is transitive in order.

8.[2p] In building ℚ from pairs (a,b) with (a,b)∼(c,d) when ad=bc, why must the pair (0,0) be excluded?

9.[3p] Which of these rules are well defined?

1.[3p]
Which of these relations on $ℤ$ are equivalence relations?

2.[2p]
Match each relation on $ℝ$ to the property it lacks.

3.[2p]
What is wrong with the argument "if $a \sim b$ then $b \sim a$ by symmetry, so $a \sim a$ by transitivity, hence every symmetric and transitive relation is reflexive"?

4.[1p]
How many relations are there on a set with $3$ elements?

5.[2p]
How many equivalence relations are there on the set ${1, 2, 3, 4}$ ?

6.[1p]
Which set is the congruence class $[2]$ modulo $3$ ?

7.[2p]
Put the lines of the proof that the relation $(a, b) \sim (c, d)$ when $a d = b c$ , on pairs with nonzero second coordinate, is transitive in order.

8.[2p]
In building $ℚ$ from pairs $(a, b)$ with $(a, b) \sim (c, d)$ when $a d = b c$ , why must the pair $(0, 0)$ be excluded?

9.[3p]
Which of these rules are well defined?