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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 to L2}. Show that R is an Equivalence Relation. Find the Set of All Lines Related to the Line Y = 2x + 4. - CBSE (Science) Class 12 - Mathematics

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Question

Let L be the set of all lines in XY plane and R be the relation in L defined as R = {(L1L2): L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y = 2x + 4.

Solution

R = {(L1L2): L1 is parallel to L2}

R is reflexive as any line L1 is parallel to itself i.e., (L1L1) ∈ R.

Now,

Let (L1L2) ∈ R.

⇒ L1 is parallel to L2.

⇒ L2 is parallel to L1.

⇒ (L2L1) ∈ R

∴ R is symmetric.

Now,

Let (L1L2), (L2L3) ∈R.

⇒ L1 is parallel to L2. Also, L2 is parallel to L3.

⇒ L1 is parallel to L3.

∴R is transitive.

Hence, R is an equivalence relation.

The set of all lines related to the line y = 2x + 4 is the set of all lines that are parallel to the line y = 2x+ 4.

Slope of line y = 2x + 4 is m = 2

It is known that parallel lines have the same slopes.

The line parallel to the given line is of the form y = 2x + c, where c ∈R.

Hence, the set of all lines related to the given line is given by y = 2x + c, where c ∈ R.

  Is there an error in this question or solution?

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 NCERT Solution for Mathematics Textbook for Class 12 (2018 to Current)
Chapter 1: Relations and Functions
Q: 14 | Page no. 6
Solution 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 to L2}. Show that R is an Equivalence Relation. Find the Set of All Lines Related to the Line Y = 2x + 4. Concept: Types of Relations.
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