Question

Write the negation of the following statements:
(i)
p:
For every real number
x
,
x
2
>
x.
(ii)
q:
There exists a rational number
x
such that
x
2
= 2.
(iii)
r:
All birds have wings.
(iv)
s:
All students study mathematics at the elementary level.

Answer 1

Negation of a statement
\[\huge \lnot(\forall x \quad f(x))=\exists x\lnot f(x)\]
A negation turns a universal quantifier into an existential, and negates whatever was inside the quantifier. Similarly...
\[\huge \lnot(\exists x \quad f(x))=\forall x \lnot f(x)\]
A negation turns an existential quantifier into a universal, and negates whatever was inside.

Answer 2

Now, if
\[\huge f(x) = x\rightarrow r\]
then
\[\huge \lnot f(x) = x \land \lnot r\]

Answer 3

(in English, the negation of the statement "if p then q" is
p and not q.)

Answer 4

Thanks

Answer 5

That conditional negation may not have been necessary :D