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Determine Whether Each of the Following Relations Are Reflexive, Symmetric and Transitive: - CBSE (Science) Class 12 - Mathematics

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Question

Determine whether each of the following relations are reflexive, symmetric and transitive:

Relation R in the set Z of all integers defined as

R = {(xy): x − y is as integer}

Solution

R = {(xy): x − y is an integer}

Now, for every x ∈ Z, (xx) ∈R as x − x = 0 is an integer.

∴R is reflexive.

Now, for every xy ∈ Z if (xy) ∈ R, then x − y is an integer.

⇒ −(x − y) is also an integer.

⇒ (y − x) is an integer.

∴ (yx) ∈ R

∴R is symmetric.

Now,

Let (xy) and (yz) ∈R, where xyz ∈ Z.

⇒ (x − y) and (y − z) are integers.

⇒ − z = (x − y) + (y − z) is an integer.

∴ (xz) ∈R

∴R is transitive.

Hence, R is reflexive, symmetric, and transitive.

  Is there an error in this question or solution?

APPEARS IN

 NCERT Solution for Mathematics Textbook for Class 12 (2018 to Current)
Chapter 1: Relations and Functions
Q: 1.4 | Page no. 5
Solution Determine Whether Each of the Following Relations Are Reflexive, Symmetric and Transitive: Concept: Types of Relations.
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