Show that the Relation R in the Set R of Real Numbers, Defined as R = {(A, B): A ≤ B2 Neither Reflexive Nor Symmetric Nor Transitive. - Mathematics

Show that the relation R in the set of real numbers, defined as

R = {(ab): a ≤ b2} is neither reflexive nor symmetric nor transitive.

Solution

R = {(ab): a ≤ b2}

It can be observed that (1/2, 1/2) in R, Since 1/2 > (1/2)^2 = 1/4

∴R is not reflexive.

Now, (1, 4) ∈ R as 1 < 42

But, 4 is not less than 12.

∴(4, 1) ∉ R

∴R is not symmetric.

Now,

(3, 2), (2, 1.5) ∈ R

(as 3 < 22 = 4 and 2 < (1.5)2 = 2.25)

But, 3 > (1.5)2 = 2.25

∴(3, 1.5) ∉ R

∴ R is not transitive.

Hence, R is neither reflexive, nor symmetric, nor transitive.

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APPEARS IN

NCERT Class 12 Maths
Chapter 1 Relations and Functions
Q 2 | Page 5