Statements (propositions) | Grade 10

"Our school team will win on Saturday." "Every prime number is odd." " $2+2=5$ ." Mathematics, like a good argument, is built from sentences that can be judged: either they are right or they are wrong. This lesson gives you the language for saying exactly what you mean, and for checking whether it is true.

What is a statement?

A sentence like (d), which contains a variable and becomes a statement once the variable is given a value, is called a statement containing variables (in Vietnamese, mệnh đề chứa biến). We write it $P(x)$ : for example $P(x)$ : " $x+1=5$ ". Then $P(4)$ is true and $P(0)$ is false.

Negation

The negation of a statement $P$ , written $\overline{P}$ , says the opposite. It is true exactly when $P$ is false, and false exactly when $P$ is true.

$P$	$\overline{P}$
true	false
false	true

Implication and equivalence

Think of a promise: "If you pass the exam, I will buy you a phone." We write it $P \Rightarrow Q$ with $P$ : "you pass", $Q$ : "I buy you a phone".

Definition Implication

The statement "if $P$ then $Q$ ", written $P \Rightarrow Q$ , is false only when $P$ is true and $Q$ is false. In every other case it is true.

$P$	$Q$	$P \Rightarrow Q$
true	true	true
true	false	false
false	true	true
false	false	true

The promise is broken only if you pass and get no phone. If you fail, the promise says nothing, so it is not broken.

The converse of $P \Rightarrow Q$ is $Q \Rightarrow P$ . A true implication can have a false converse.

For example, "a triangle has three equal sides $\Leftrightarrow$ it has three equal angles" is true.

Necessary and sufficient conditions

When $P \Rightarrow Q$ is true we say:

$P$ is a sufficient condition for $Q$ (knowing $P$ is enough to be sure of $Q$ );
$Q$ is a necessary condition for $P$ (if $Q$ fails, $P$ cannot hold).

When $P \Leftrightarrow Q$ is true, each is a necessary and sufficient condition for the other.

Quantifiers: "for all" and "there exists"

$\forall$ means "for all" (for every). The statement $\forall x \in X,\ P(x)$ says that $P(x)$ holds for every $x$ in $X$ .
$\exists$ means "there exists" (at least one). The statement $\exists x \in X,\ P(x)$ says that at least one $x$ in $X$ makes $P(x)$ true.

Negating quantifiers

To show "everything has property $P$ " is wrong you need just one thing without it. To show "something has property $P$ " is wrong you must show that nothing has it. Hence:

Disproving by a counter-example

To prove that a statement " $\forall x,\ P(x)$ " is false, it is enough to give one value of $x$ for which $P(x)$ is false. Such a value is a counter-example. (Examples never prove a "for all" statement, but one counter-example disproves it.)

Key terms

English	Tiếng Việt
statement	mệnh đề
negation	mệnh đề phủ định
implication	mệnh đề kéo theo
converse	mệnh đề đảo
equivalence	mệnh đề tương đương
necessary condition	điều kiện cần
sufficient condition	điều kiện đủ
quantifier	kí hiệu lượng từ
counter-example	phản ví dụ

Summary

A statement is a sentence that is true or false, never both; a sentence with a variable becomes a statement when the variable gets a value.
$\overline{P}$ has the opposite truth value of $P$ . $P \Rightarrow Q$ is false only when $P$ is true and $Q$ is false.
$P \Leftrightarrow Q$ means $P \Rightarrow Q$ and $Q \Rightarrow P$ . If $P \Rightarrow Q$ : $P$ is sufficient for $Q$ , $Q$ is necessary for $P$ .
The negation of $\forall$ is $\exists$ and the negation of $\exists$ is $\forall$ , with the property negated.
One counter-example disproves a "for all" statement.

Practice

Which of these are statements? If so, are they true or false? (a) $15$ is divisible by $4$ . (b) $x+2>0$ . (c) Please open the door. (d) $2^3=8$ .
Write the negation of (a) " $17$ is a prime number", (b) " $5 \ge 3$ ", (c) " $\sqrt{2}$ is rational". Which is true, the statement or its negation?
Let $P$ : " $x=3$ " and $Q$ : " $x^2=9$ ". Write $Q \Rightarrow P$ in words and decide whether it is true.
State the converse of "If a number ends in $0$ then it is divisible by $5$ ." Is the original true? Is the converse true?
Fill in "necessary", "sufficient" or "necessary and sufficient": (a) "ABCD is a square" is a ... condition for "ABCD is a rectangle". (b) " $a=0$ " is a ... condition for " $ab=0$ ". (c) " $x$ is even" is a ... condition for " $x+1$ is odd".
Decide whether each is true: (a) $\forall x\in\mathbb{R},\ x^2\ge 0$ . (b) $\exists x\in\mathbb{R},\ x^2=-1$ . (c) $\exists n\in\mathbb{N},\ n^2=n$ . (d) $\forall n\in\mathbb{N},\ n^2>n$ .
Negate " $\exists x\in\mathbb{R},\ x^2-4x+4<0$ " and say which of the two statements is true.
Use a counter-example to disprove: (a) "Every prime number is odd." (b) "If $x^2>1$ then $x>1$ ."
Is " $x^2=4 \Leftrightarrow x=2$ " true? If not, correct it by changing one side so that it becomes a true equivalence.
Challenge: for a natural number $n$ , show that " $n$ is even $\Leftrightarrow$ $n^2$ is even" is true.

Show solutions

(a) A statement, false ( $15=4\times 3+3$ ). (b) Not a statement: it depends on $x$ . (c) Not a statement: it is a request. (d) A statement, true.
(a) " $17$ is not a prime number"; the statement is true, the negation false. (b) " $5<3$ "; the statement is true, the negation false. (c) " $\sqrt{2}$ is not rational"; the statement is false, the negation true.
$Q \Rightarrow P$ : "If $x^2=9$ then $x=3$ ." It is false: $x=-3$ gives $x^2=9$ but $x\ne 3$ .
Converse: "If a number is divisible by $5$ then it ends in $0$ ." The original is true ( $10k=5\cdot 2k$ ). The converse is false: $15$ is a counter-example.
(a) "ABCD is a square" $\Rightarrow$ "ABCD is a rectangle" is true, the converse is false, so sufficient. (b) $a=0 \Rightarrow ab=0$ is true; $ab=0 \Rightarrow a=0$ is false (take $b=0$ , $a=1$ ), so sufficient. (c) $x$ even $\Rightarrow x+1$ odd, and $x+1$ odd $\Rightarrow x$ even, so necessary and sufficient.
(a) True. (b) False. (c) True: $n=0$ (or $n=1$ ) gives $n^2=n$ . (d) False: $n=1$ gives $1>1$ , which is wrong.
Negation: " $\forall x\in\mathbb{R},\ x^2-4x+4\ge 0$ ". Since $x^2-4x+4=(x-2)^2\ge 0$ for all $x$ , the negation is true and the original is false.
(a) $2$ is prime and even. (b) $x=-2$ : $x^2=4>1$ but $x=-2$ is not greater than $1$ .
It is false: " $\Leftarrow$ " holds ( $x=2\Rightarrow x^2=4$ ) but " $\Rightarrow$ " fails ( $x=-2$ ). A true version: " $x^2=4 \Leftrightarrow x=2$ or $x=-2$ ", or " $x^2=4$ and $x>0 \Leftrightarrow x=2$ ".
( $\Rightarrow$ ) If $n=2k$ then $n^2=4k^2=2\cdot 2k^2$ is even. ( $\Leftarrow$ ) Suppose $n$ is odd, $n=2k+1$ . Then $n^2=4k^2+4k+1=2(2k^2+2k)+1$ is odd. So if $n^2$ is even, $n$ cannot be odd, hence $n$ is even. Both directions hold, so the equivalence is true.