`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."`
`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."`
