Mathematical Language

Read this section for an introduction to mathematical language, then work through practice problems 1-4.

For some simple statements we can construct the negation just by adding the word "not".

Statement	Negation of the Statement
$x$ is equal to $3 ( x = 3 )$	$x$ is not equal to $3 ( x ≠ 3 )$
$x$ is less than $5 ( x < 5 )$	$x$ is not less than $5 ( x$ $x$ is greater than or equal to $5 ( x ≥ 5 )$

When the statement contains words such as "all", "no", or "some," then its negation is more complicated.

Statement	Negation of the Statement
All $x$ satisfy $A$ . Every $x$ satisfies $A$ .	At least one $x$ does not satisfy $A$ . There is an $x$ which does not satisfy $A$ . Some $x$ does not satisfy $A$ .
No $x$ satisfies $A$ . Every $x$ does not satisfy $A$ .	At least one $x$ satisfies $A$ . Some $x$ satisfies $A$ .
There is an $x$ which satisfies $A$ . At least one $x$ satisfies $A$ . Some $x$ satisfies $A$ .	No $x$ satisfies $A$ . Every $x$ does not satisfy $A$ .

We can also negate compound statements containing "and" and "or".

Statement	Negation of the Statement
$A$ and $B$ are both true.	At least one of $A$ or $B$ is not true.
$A$ and $B$ and $C$ are all true.	At least one of $A$ or $B$ or $C$ is not true.
$A$ or $B$ is true.	Both $A$ and $B$ are not true.

Practice 2: Write the negation of each of these statements.

(a)

$x + 5 ≥ 3$

(b) All prime numbers are odd.

(c)

$x^2 < 4$

(d)

$x$ divides 2 and

$x$ divides 3.

(e) No mathematician can sing well.