Question

The relation is less than given by {(a, b) /a<b} on R is a) reflexive b) symmetric c) transitive d) equivalent​

Answer 1

Answer:

Hard

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Answer

R={(a,b):a≤b

3

}

Here R is set of real numbers

Hence both a and b are real numbers.

1.If the relation is reflexive, then (a,a)∈R

i.e.,a≤a

3

For a=1,a

3

≤1⇒1≤1

For a=2,a

3

≤8⇒2≤8

For a=

2

1

,a

3

≥

8

1

Hence a≤a

3

is not true for all values of a

So, the given relation is not relexive.

To check whether symmetric or not:

If (a,b)∈R then (b,a)∈R

If a≤b

3

then b≤a

3

For a=2,b=3,2<2

3

,3<2

3

For a=2,b=9,2<9

3

,9>2

3

Since b≤a

3

is not true for all values of a and b

Hence the given relation is not symmetric.

To check whether transitive or not:

If (a,b)∈R and (b,c)∈R then (a,c)∈R

If a≤b

3

and b≤c

3

then a≤c

3

For a=1,b=2,c=3,b

3

=8,c

3

=27⇒a≤b

3

,b≤c

3

and a≤c

3

For a=3,b=

2

3

,c=

3

4

,b

3

=(

2

3

)

3

=3.375,c

3

=(

3

4

)

3

=2.37⇒a≤b

3

,b≤c

3

and a≥c

3

Since if a≤b

3

,b≤c

3

and a≤c

3

is not true for all values of a,b,c.

Hence, the given relation is not transitive.

∴ the given relation is neither reflexive, symmetric or transitive.