Let L be the set of all lines in XY plane and R be the relation in L defined as $R = \{(L_1, L_2) : L_1$ is parallel to $L_2\}$. Show that R is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4$.

Let L be the set of all lines in XY plane and R be the relation in L defined as R = \(L₁, L₂) : L₁ is parallel

Exercise 1.1 · Q14

R = \{(L_1, L_2) : L_1

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The relation "is parallel to" on the set of all lines in the plane is an equivalence relation because it is reflexive (a line is parallel to itself), symmetric (if one line is parallel to another, the other is parallel to the first), and transitive (if line A is parallel to B and B is parallel to C, then A is parallel to C). The set of all lines related to $y = 2x + 4$ is exactly the set of all lines with slope $2$ , i.e., all lines of the form $y = 2x + c$ where $c \in \mathbb{R}$ .

Why this works: The idea behind equivalence relations

An equivalence relation is a way of grouping objects that are "the same" in some precise sense. For lines in the plane, being parallel is a natural way to group them — all lines that point in the same direction belong together. The three properties (reflexive, symmetric, transitive) are what make this grouping well-behaved: every line is in its own group, the grouping doesn't depend on which line you pick first, and if two lines are both parallel to a third, they must be in the same group.

The key geometric fact: parallel lines have the same slope. So the equivalence class of a given line is simply all lines with that same slope.

Watch out

A common mistake is to forget that a line is considered parallel to itself. In everyday language, "parallel" often means "distinct and never meeting," but in mathematics, the definition includes the case of a line being parallel to itself. This is essential for reflexivity.

Step-by-step proof

1. Reflexive property

We need to show that every line $L_1$ is related to itself, i.e., $(L_1, L_1) \in R$ .

By definition, $R$ contains $(L_1, L_2)$ if $L_1$ is parallel to $L_2$ . Is a line parallel to itself? Yes — a line has the same direction as itself, and in formal geometry, parallelism includes the case of identity. So $L_1 \parallel L_1$ holds for every line $L_1$ .

Thus $R$ is reflexive.

2. Symmetric property

We need to show: if $(L_1, L_2) \in R$ , then $(L_2, L_1) \in R$ .

If $L_1$ is parallel to $L_2$ , then by the definition of parallelism, $L_2$ is also parallel to $L_1$ — it's a mutual relationship. If two lines have the same slope, it doesn't matter which one you name first. So $(L_2, L_1)$ is also in $R$ .

Thus $R$ is symmetric.

3. Transitive property

We need to show: if $(L_1, L_2) \in R$ and $(L_2, L_3) \in R$ , then $(L_1, L_3) \in R$ . …

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Same / Similar Concept — real previous-year questions on the same or a closely similar concept, not this exact question.

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CBSE 2026Set ANNUAL1 mark
$Q. Write the answer in one word/sentence: Define equivalence relation.$
›Reveal solutionSolution

Equivalence relation = reflexive + symmetric + transitive.

…

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CBSE 2025Set ANNUAL1 mark
$R = \{(1,-1),(2,-2),(3,-1)\}$
›Reveal solutionSolution
State the three defining properties of an equivalence relation.

A relation $R$ on a non-empty set $A$ is called an equivalence relation if it satisfies all three of the following:

Reflexive: $(a,a)\in R$ for every $a\in A$ .

Symmetric: if $(a,b)\in R$ then $(b,a)\in R$ .

Transitive: if $(a,b)\in R$ and $(b,c)\in R$ then $(a,c)\in R$ . …
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CBSE 2024Set A1 mark
$R$
›Reveal solutionSolution

A relation that is reflexive, symmetric and transitive is called an equivalence relation.

…

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CBSE 2024Set ANNUAL1 markMCQ
$Q. Let R be the relation in the set Z of all integers defined as, R = {(x, y) : x - y is an integer}, then R is$
(a) Reflexive
(b) Symmetric
(c) Transitive
(d) Equivalence relation
›Reveal solutionSolution

R is reflexive, symmetric, and transitive, hence an equivalence relation.

For any $x \in \mathbb{Z}$ , $x - x = 0$ , which is an integer, so $(x,x) \in R$ for every $x$ . Hence R is reflexive.

If $(x,y) \in R$ , then $x-y$ is an integer. Then $y - x = -(x-y)$ is also an integer, so $(y,x) \in R$ . Hence R is symmetric.

…

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CBSE 2023Set ANNUAL1 mark
$R$
›Reveal solutionSolution

For an equivalence relation R on A, both the domain and the range of R equal A itself.

Since R is an equivalence relation on A, it is reflexive: for every $a \in A$ , $(a,a) \in R$ . This means every element of $A$ appears as a first coordinate (so it is in the domain) and also as a second coordinate (so it is in the range) of some pair in R. H …

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CBSE 2023Set ANNUAL1 mark
$R$
›Reveal solutionSolution
An equivalence relation is one that is simultaneously reflexive, symmetric and transitive; the alternative just lists the images of the allowed $x$ -values.

A relation $R$ on a set $A$ is called an equivalence relation when it satisfies all three of the following properties:

Reflexive: $(a,a)\in R$ for every $a\in A$ .

Symmetric: if $(a,b)\in R$ then $(b,a)\in R$ .

Transitive: if $(a,b)\in R$ and $(b,c)\in R$ then $(a,c)\in R$ . …
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CBSE 2023Set ANNUAL1 mark
$Q. Define an equivalence relation.$
›Reveal solutionSolution
An equivalence relation is a relation satisfying all three properties: reflexivity, symmetry, and transitivity.

A relation $R$ defined on a set $A$ is called an equivalence relation if it satisfies all of the following three conditions:

Reflexive: $(a,a) \in R$ for every $a \in A$ .

Symmetric: If $(a,b) \in R$ , then $(b,a) \in R$ , for all $a,b \in A$ .

Transitive: If $(a,b)\in R$ and $(b,c)\in R$ , then $(a,c)\in R$ , for all $a,b,c \in A$ . …
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CBSE 2023Set ANNUAL1 markMCQ
$B=\{b_1,b_2,b_3\}$
(a) $2^{6}$
(b) $2^{5}$
(c) $0$
(d) $2^{3}$
›Reveal solutionSolution

The number of relations from a set of size $m$ to a set of size $n$ is $2^{mn}$ ; here $m=3,n=2$ .

$B=\{b_1,b_2,b_3\}$ has $|B|=3$ elements; $G=\{g_1,g_2\}$ has $|G|=2$ elements.

A relation from $B$ to $G$ is any subset of $B\times G$ . The number of elements in $B\times G$ is $|B|\times|G|=3\times2=6$ .

…

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CBSE 2022Set ANNUAL1 markMCQ
$Q. Which of the following relations on A = {1, 2, 3} is an equivalence relation?$
(a) {(1, 1), (2, 2), (3, 3)}
(b) {(1, 1), (2, 2), (3, 3), (1, 2)}
(c) {(1, 1), (3, 3), (1, 3), (3, 1)}
(d) None of these
›Reveal solutionSolution

Only option (a) is reflexive, symmetric, and transitive at once - the other two options fail reflexivity or symmetry.

An equivalence relation on $A=\{1,2,3\}$ must be reflexive (contain $(1,1),(2,2),(3,3)$ ), symmetric (if $(x,y)$ is in it so is $(y,x)$ ), and transitive.

(a) $\{(1,1),(2,2),(3,3)\}$ - contains all three diagonal pairs (reflexive), has no off-diagonal pair to break symmetry, and is trivially transitive. This IS an equivalence relation (it is the identity relation on $A$ ).

…

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CBSE 2022Set ANNUAL1 markMCQ
$y=x-4$
(a) Equivalence
(b) Only reflexive
(c) Not reflexive
(d) symmetric but not transitive
›Reveal solutionSolution

The relation 'is parallel to' on a set of lines is reflexive, symmetric and transitive, hence an equivalence relation.

$R=\{(L_1,L_2): L_1\parallel L_2\}$ on the set $L$ of all lines.

Reflexive: every line is parallel to itself, so $(L,L)\in R$ for all $L$ . (True)

Symmetric: if $L_1\parallel L_2$ then $L_2\parallel L_1$ , so $(L_1,L_2)\in R\Rightarrow(L_2,L_1)\in R$ . (True)

…

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CBSE 2020Set ANNUAL1 mark
$Q. What is meant by an equivalence relation?$
›Reveal solutionSolution

definition recall

…

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CBSE 2019Set ANNUAL1 markMCQ
$R\{(a, b) : a$
(a) No relation
(b) Trivial
(c) Equivalence relation
(d) Not symmetric
›Reveal solutionSolution

“Same parity” partitions A into evens and odds, so R is reflexive, symmetric and transitive — an equivalence relation.

R = {(a, b) : a and b are both even or both odd} on A = {1,2,3,4,5,6,7}.

Step 1 (Reflexive): Any a has the same parity as itself, so (a, a) ∈ R. ✓

Step 2 (Symmetric): If a and b have the same parity, so do b and a; (a,b) ∈ R ⇒ (b,a) ∈ R. ✓

…

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Q.Let L be the set of all lines in XY plane and R be the relation in L defined as $R = \{(L_1, L_2) : L_1$ is parallel to $L_2\}$ . Show that R is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4$ .

Concept understanding — Equivalence Relation Proof

Proving a Relation is an Equivalence Relation

The three checks

A worked template

Why this works: The idea behind equivalence relations

Step-by-step proof

1. Reflexive property

2. Symmetric property

3. Transitive property

Method: Proving a geometric "sameness" relation is an equivalence relation

Steps

Common Mistakes

Q.Let L be the set of all lines in XY plane and R be the relation in L defined as R={(L1,L2):L1R = \{(L_1, L_2) : L_1R={(L1​,L2​):L1​ is parallel to L2}L_2\}L2​}. Show that R is an equivalence relation. Find the set of all lines related to the line y=2x+4y = 2x + 4y=2x+4.

Why this works: The idea behind equivalence relations

Step-by-step proof

1. Reflexive property

2. Symmetric property

3. Transitive property

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Q.Let L be the set of all lines in XY plane and R be the relation in L defined as $R = \{(L_1, L_2) : L_1$ is parallel to $L_2\}$ . Show that R is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4$ .

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