Mathematics MATHEMATICAL REASONING - Statements
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star Statements

### Statements

color{green} ✍️ The basic unit involved in mathematical reasoning is a mathematical statement.

color{green} ✍️ Let us start with two sentences:

 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \color{orange} "In 2003, the president of India was a woman."

 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \color{orange} { "An elephant weighs more than a human being."

When we read these sentences, we immediately decide that the first sentence is false and the second is correct. There is no confusion regarding these.

In mathematics such sentences are color(blue)"called statements."

color{green} ✍️ On the other hand, consider the sentence: color(orange)"Women are more intelligent than men."

Some people may think it is true while others may disagree. Regarding this sentence we cannot say whether it is always true or false .

That means this sentence is color(green)("ambiguous").

Such a sentence is color(purple)("not acceptable as a statement in mathematics.")

\color{green} ✍️ \color{green} \mathbf(KEY \ CONCEPT)

color(blue)"A sentence is called a mathematically acceptable statement if it is either true or false but not both."

Whenever we mention a statement here, it is a “mathematically acceptable” statement.

color{green} ✍️ While studying mathematics, we come across many such sentences.

color(red)("Some examples are :")

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ "Two plus two equals four. "

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ "The sum of two positive numbers is positive."

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ "All prime numbers are odd numbers."

Of these sentences, the first two are true and the third one is false. There is color(green)("no ambiguity") regarding these sentences. Therefore, color(green)("they are statements").

color(red)("Now, consider the following sentences :")

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ "How beautiful !"

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ "Open the door."

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ "Where are you going?"

Are they statements? No, because the first one is an exclamation, the second an order and the third a question.
color(green)("None of these is considered as a statement in mathematical language. ")

color{green} ✍️ Sentences involving variable time such as color(red)("“today”, “tomorrow”") or color(red)("“yesterday”") are not statements. This is because it is not known what time is referred here.

color(red)("For example : ")

The sentence "Tomorrow is Friday" is not a statement. The sentence is correct (true) on a Thursday but not on other days.

color{green} ✍️ The same argument holds for sentences with pronouns unless a particular person is referred to and for variable places such as color(blue)("“here”, “there”") etc.,

color(red)("For example, the sentences")

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \color(orange) "She is a mathematics graduate."

\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \color(orange) "Kashmir is far from here." color(blue)("are not statements.")

color{green} ✍️ color(red)("Here is another sentence")

" "color(orange)"There are 40 days in a month."

Note that the period mentioned in the sentence above is a “variable time” that is any of 12 months. But we know that the sentence is always false (irrespective of the month) since the maximum number of days in a month can never exceed 31.

Therefore,color(blue)("this sentence is a statement.") So, what makes a sentence a statement is the fact that the sentence is either true or false but not both.

color{green} ✍️ While dealing with statements, we usually denote them by small letters p, q, r,...

color(red)("For example :") we denote the statement color(orange)"“Fire is always hot”" by p.

This is also written as color(green)"p: Fire is always hot."
Q 3131601522

Check whether the following sentences are statements. Give reasons for your answer.

(i) 8 is less than 6.
(ii) Every set is a finite set.
(iii) The sun is a star.
(iv) Mathematics is fun.
(v) There is no rain without clouds.
(vi) How far is Chennai from here?

Solution:

(i) This sentence is false because 8 is greater than 6. Hence it is a statement.
(ii) This sentence is also false since there are sets which are not finite. Hence it is a statement.
(iii) It is a scientifically established fact that sun is a star and, therefore, this sentence is always true. Hence it is a statement.
(iv) This sentence is subjective in the sense that for those who like mathematics, it may be fun but for others it may not be. This means that this sentence is not always true. Hence it is not a statement.
(v) It is a scientifically established natural phenomenon that cloud is formed before it rains. Therefore, this sentence is always true. Hence it is a statement.
(vi) This is a question which also contains the word “Here”. Hence it is not a statement. The above examples show that whenever we say that a sentence is a statement we should always say why it is so. This “why” of it is more important than the answer.