MATHEMATICAL REASONING George Boole ( - ) MATHEMATICS 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 called statements . On the other hand, consider the sentence: 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 ambiguous. Such a sentence is not acceptable as a statement in mathematics.
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. While studying mathematics, we come across many such sentences. 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 no ambiguity regarding these sentences.
Therefore, they are statements. Can you think of an example of a sentence which is vague or ambiguous? Consider the sentence: The sum of x and y is greater than Here, we are not in a position to determine whether it is true or false, unless we know what x and y are. For example, it is false where x = , y = – and true when x = and y = .
Therefore, this sentence is not a statement. But the sentence: For any natural numbers x and y, the sum of x and y is greater than is a statement. 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. None of these is considered as a statement in mathematical language.
Sentences involving variable time such as “today”, “tomorrow” or “yesterday” are not statements. This is