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.

KCET 2025Set A-11 markMCQ
$A = \{a, b, c\}$
›Reveal solutionSolution
Equivalence relations = partitions; requiring $(b,c)$ puts $b, c$ in one block, leaving only two possible partitions.

Step 1 — The key principle. Every equivalence relation on a set is reflexive, symmetric and transitive, and there is a one-to-one correspondence between equivalence relations on $A$ and partitions of $A$ (the blocks are the equivalence classes). So instead of listing relations, count partitions.

Step 2 — Impose the condition. The relation must contain $(b, c)$ . Hence $b \sim c$ , so $b$ and $c$ must lie in the same equivalence class.

Step 3 — Enumerate the partitions of $\{a,b,c\}$ in which $b, c$ share a block.

$\{\{a\},\ \{b, c\}\}$ — gives the relation $R_1 = \{(a,a),(b,b),(c,c),(b,c),(c,b)\}.$ …
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COMEDK 2024Set 2024-E1 markMCQ
$\mathrm{R}$
›Reveal solutionSolution
An equivalence relation must be reflexive, symmetric, and transitive. Only option (A) satisfies all three properties on the real numbers.

Concept & Intuition

An equivalence relation is a way of saying two things are "the same" in some abstract sense. To be a proper notion of sameness, it must obey three rules:

Reflexive: Everything is the same as itself.

Symmetric: If A is the same as B, then B is the same as A.

Transitive: If A is the same as B and B is the same as C, then A is the same as C.

We test each given relation against these three properties. If any one fails, it is not an equivalence relation.

Step-by-step analysis

Option (A): $a R_1 b \iff |a| = |b|$

Reflexive: For any real $a$ , $|a| = |a|$ always holds. ✓

Symmetric: If $|a| = |b|$ , then clearly $|b| = |a|$ . ✓

Transitive: If $|a| = |b|$ and $|b| = |c|$ , then $|a| = |c|$ . ✓ All three properties hold, so this is an equivalence relation. The equivalence classes are sets of numbers with the same absolute value (e.g., $\{2, -2\}$ ).

Option (B): $a R_3 b \iff a \text{ divides } b$ (on real numbers)

Reflexive: Does $a$ divide $a$ ? For real numbers, "divides" usually means $b/a$ is an integer. For $a=0$ , division is undefined; for $a \neq 0$ , $a/a = 1$ is an integer, so it holds for nonzero $a$ . But reflexivity must hold for every real number. Since $0$ does not divide itself (division by zero is undefined), reflexivity fails. ✗ (Even if we restrict to nonzero reals, symmetry fails: $2$ divides $4$ but $4$ does not divide $2$ .)

Option (C): $a R_2 b \iff a \geq b$

Reflexive: $a \geq a$ is true for all $a$ . ✓ …
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COMEDK 2023Set 2023-E1 markMCQ
$\mathrm{W}$
›Reveal solutionSolution

Each listed element leaves remainder 1: 1 = 5(0)+1, 6 = 5(1)+1, 11 = 5(2)+1, ... (The other options are the classes of 2, 4 and 0 respectively.)

Concept: equivalence classes of the relation 'congruent modulo 5' on the whole numbers.

aRb iff a and b leave the same remainder on division by 5, i.e. a = b (mod 5).

The equivalence class of 1 is

[1] = { w in W : w = 1 (mod 5) } = { 1, 6, 11, 16, 21, ... }. …
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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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