# Relations and Equivalence

Mathematics I · Mathematical Language and Reasoning · https://tryals.app/learn/mathematics-i/relations-and-equivalence

## Binary Relations

A **binary relation** $R$ on a set $A$ is a set of ordered pairs, recording which elements are connected. We write $a\, R\, b$ when $(a, b) \in R$. Relations generalize functions to model any pairwise connection.

Three fundamental properties classify relations:

| Property | Definition | Example ($\le$ on $\mathbb{R}$) |
| --- | --- | --- |
| **Reflexive** | $a\, R\, a$ for every $a$ | $a \le a$ (True) |
| **Symmetric** | $a\, R\, b \Rightarrow b\, R\, a$ | $a \le b \Rightarrow b \le a$ (False) |
| **Transitive** | $a\, R\, b$ and $b\, R\, c \Rightarrow a\, R\, c$ | $a \le b$ and $b \le c \Rightarrow a \le c$ |

A relation with all three is an **equivalence relation**, capturing sameness in some respect.

## Structure & Pitfalls

An **equivalence relation** **partitions** set $A$ into disjoint **equivalence classes** of mutually related elements. For example, congruence mod 5 partitions integers into five remainder classes.

A different mix—reflexive, **antisymmetric** ($a\,R\,b$ and $b\,R\,a \Rightarrow a = b$), and transitive—forms a **partial order** like $\le$.

> **Common pitfall:** Confusing **symmetric** with **transitive**. Symmetry reverses a single pair ($a\,R\,b \Rightarrow b\,R\,a$); transitivity chains two pairs ($a\,R\,b$ and $b\,R\,c \Rightarrow a\,R\,c$). "One step from" on a number line is symmetric but not transitive.

## Practice questions

8 of this lesson's 13 practice questions, with answers. The full set is in the app.

### 1. Complete the definition of transitivity.

**Answer:** A relation is transitive when, whenever a relates to b and b relates to c, it follows that **a relates to c**.

**Why:** Transitivity says the relation composes: two links a-b and b-c collapse into a single link a-c. 'b relates to a' is symmetry, and 'a relates to a' is reflexivity, three distinct requirements.

Page: https://tryals.app/practice/mathematics-i/relations-and-equivalence/complete-the-definition-of-transitivity

### 2. Let R be 'has the same parity as' on {1, 2, 3, 4}. Select every pair that belongs to R.

A. (4, 1)
B. (2, 4)
C. (1, 2)
D. (1, 3)
E. (3, 3)

**Answer:** B. (2, 4); D. (1, 3); E. (3, 3)

**Why:** (1,3) are both odd, (2,4) both even, and (3,3) is reflexive. (1,2) and (4,1) mix an odd with an even, so they are excluded. The relation splits the set into {1,3} and {2,4}, its two equivalence classes.

Page: https://tryals.app/practice/mathematics-i/relations-and-equivalence/let-r-be-has-the-same-parity-as-on-1-2-3-4-select-every-pair

### 3. "Has the same remainder mod 3" partitions the integers into three classes. Why is the quotient construction, treating each class as a single object, one of the most reused moves in mathematics?

A. Because it is the only way to define arithmetic on infinite sets
B. Because it reduces infinite sets to finite ones, which are easier to study
C. Because it builds a smaller structure where the distinction has gone
D. Because equivalence classes are always easier to compute with than their elements

**Answer:** C. Because it builds a smaller structure where the distinction has gone

**Why:** You declare a distinction irrelevant and build a world where it no longer exists, and the operations survive, because addition was well defined on classes. A clock face is ℤ/12ℤ, and the same move constructs the rationals from pairs of integers and quotient groups throughout algebra.

Page: https://tryals.app/practice/mathematics-i/relations-and-equivalence/has-the-same-remainder-mod-3-partitions-the-integers-into-three

### 4. Order the steps that verify 'has the same parity as' is an equivalence relation.

**Answer:**

1. Reflexive: every integer has the same parity as itself
2. Symmetric: if a and b share a parity, then b and a do too
3. Transitive: if a matches b and b matches c, then a matches c
4. All three properties hold, so the relation is an equivalence relation

**Why:** Verification is mechanical: check reflexivity, symmetry and transitivity in turn, then conclude. Skipping any one of them leaves the conclusion unearned, and it is transitivity that most often fails in practice.

Page: https://tryals.app/practice/mathematics-i/relations-and-equivalence/order-the-steps-that-verify-has-the-same-parity-as-is-an

### 5. An equivalence relation partitions its set into classes with no overlaps and nothing left out. Why do reflexivity, symmetry and transitivity together force that structure?

A. Because reflexivity leaves nothing out, and symmetry with transitivity forbid overlap.
B. Because transitivity alone forces the classes, with reflexivity and symmetry serving only to make the relation well behaved.
C. Because the three properties are the definition of a partition, so the connection is true by stipulation.
D. Because equivalence classes are constructed to be disjoint, and the three properties then follow from that construction.

**Answer:** A. Because reflexivity leaves nothing out, and symmetry with transitivity forbid overlap.

**Why:** Reflexivity puts every element in its own class, so nothing is left out. For overlaps: if [a] and [b] share z then a ~ z and b ~ z, symmetry and transitivity give a ~ b, and the classes coincide. All three are load-bearing, 'is less than' is transitive alone and partitions nothing. The converse holds too, so partitions and equivalence relations are one idea.

Page: https://tryals.app/practice/mathematics-i/relations-and-equivalence/an-equivalence-relation-partitions-its-set-into-classes-with-no

### 6. Every function is a relation.

**Answer:** True

**Why:** True. A function is a relation with the extra restriction that each domain element appears in exactly one pair. The converse fails: 'differs by 1' relates 3 to both 2 and 4, so it is a relation but not a function.

Page: https://tryals.app/practice/mathematics-i/relations-and-equivalence/every-function-is-a-relation

### 7. An equivalence relation is reflexive, symmetric AND transitive. Sort each relation.

**Answer:**

- Equivalence relation: 'has the same remainder mod 3 as', 'equals'
- Not an equivalence relation: 'is less than or equal to', 'divides'

**Why:** Congruence mod 3 and equality satisfy all three properties. 'At most' is reflexive and transitive but not symmetric (2 <= 5 while 5 <= 2 fails), and 'divides' fails symmetry likewise (2 divides 6, but 6 does not divide 2).

Page: https://tryals.app/practice/mathematics-i/relations-and-equivalence/an-equivalence-relation-is-reflexive-symmetric-and-transitive-sort

### 8. The relation 'has the same remainder when divided by 3' partitions the integers into equivalence classes. How many classes are there?

**Answer:** 3

**Why:** The possible remainders are 0, 1 and 2, giving exactly three classes. Every integer lands in precisely one of them, the defining feature of a partition, and the reason equivalence relations and partitions are two views of the same idea.

Page: https://tryals.app/practice/mathematics-i/relations-and-equivalence/the-relation-has-the-same-remainder-when-divided-by-3-partitions