Question

Any equation or inequality with variables in it is a predicate in the domain of real numbers. For the following statement, tell whether the statement is true or false. (∀x)(x4> x)

Answer 1

Answer with explanation:

The statement is given by:

∀ x ,
`x^4>x`

This statement is false

Since, if we consider,

`x=(1)/(2)`

then we have:

`x^4=((1)/(2))^4\\\\i.e.\\\\x^4=(1)/(2^4)\\\\i.e.\\\\x^4=(1)/(16)`

Also, we know that:

`(1)/(16)<(1)/(2)`

( Since, two number with same numerator; the number with greater denominator is smaller than the number with the smaller denominator )

Hence, we get:

`x^4<x`

when
`x=(1)/(2)`

Hence, the result :

`x^4>x` is not true for all x belonging to real numbers.

Hence, the given statement is a FALSE statement.