A Note While forming the negation of a statement, phrases like, “It is not the case” or “It is false that” are also used. Here is an example to illustrate how, by looking at the negation of a statement, we may improve our understanding of it. Let us consider the statement p: Everyone in Germany speaks German. The denial of this sentence tells us that not everyone in Germany speaks German.
This does not mean that no person in Germany speaks German. It says merely that at least one person in Germany does not speak German. We shall consider more examples. Example Write the negation of the following statements.
Both the diagonals of a rectangle have the same length. is rational. Solution (i) This statement says that in a rectangle, both the diagonals have the same length. This means that if you take any rectangle, then both the diagonals have the same length.
The negation of this statement is It is false that both the diagonals in a rectangle have the same length This means the statement There is atleast one rectangle whose both diagonals do not have the same length. (ii) The negation of the statement in (ii) may also be written as It is not the case that is rational. This can also be rewritten as is not rational. MATHEMATICS Example Write the negation of the following statements and check whether the resulting statements are true, Australia is a continent.
There does not exist a quadrilateral which has all its sides equal. (iii) Every natural number is greater than . (iv) The sum of and is . Solution (i) The negation of the statement is It is false that Australia is a continent.
This can also be rewritten as Australia is not a continent. We know that this statement is false. The negation of the statement is It is not the case that there does not exist a quadrilateral which has all its sides equal. This also means the following: There exists a quadrilateral which has all its sides equal.