Question

Suppose there is a relation * between the positive numbers x and y given by x * y if and only if `xley^(2)`. Then which one of the following is correct?
A. * is reflexive but not transitive and symmetric
B. * is transitive but not reflexive and symmetric
C. * is symmetric and reflexive but not transitive
D. * is symmetric and but not reflexive and transitive

Answer 1

Correct Answer - A
x and y are positive numbers.
`x le y^(2)`
Reflexive-
`x lex^(2)AA` positive numbers.
Hence relation is reflexive.
Transitive-
`x ley^(2)" "ylez^(2)`
Let x=5, y=3, z=2
`5le(3)^(2)" "(3)le(2)^(2)`
but `5cancelle(2)^(2)`
Hence, `xley^(2)" "ylez^(2)`
`"but " x cancelle z(2)`
Thus relation is not transitive.
Symmetric
`1 le (2)^(2)" while "2cancelle(1)^(2)`
Hence relation is not symmetric.
Thus `xley^(2)AA` positive numbers is reflexive, but not transitive and symmetric.