Negation of a Statement

In general, a statement is a meaningful sentence that is not an exclamation, or question or order. A statement is acceptable only if it is either always true or always false. Sometimes in Mathematics, it is necessary to find the opposite of the given mathematical statement. The process of finding the opposite of the given statement is called “Negation”. In this article, we are going to learn the negation of a statement with many solved examples.

What is Meant by Negation of a Statement?

In Mathematics, the negation of a statement is the opposite of the given mathematical statement. If “P” is a statement, then the negation of statement P is represented by ~P. The symbols used to represent the negation of a statement are “~” or “¬”.

For example, the given sentence is “Arjun’s dog has a black tail”. Then, the negation of the given statement is “Arjun’s dog does not have a black tail”. Thus, if the given statement is true, then the negation of the given statement is false.

It is observed that the “negation of the negated sentence is the original sentence”. Now, let us understand this with the help of an example.

Assume that the given sentence, P is “Triangle ABC is an equilateral triangle”.

Thus, the negation of the given sentence, ~P is “Triangle ABC is not an equilateral triangle”.

The negation of the negated sentence ~(~P) is “Triangle ABC is an equilateral triangle”.

Hence, this proves that the negation of the negated sentence is the given original sentence.

Working Rule for Obtaining the Negation of a Statement

The working rule for obtaining the negation of a statement is given below:

1. Write the given statement with “not”.

For example, the sum of 2 and 2 is 4. The negation of the given statement is “the sum of 2 and 2 is not 4”.

2. Make suitable modifications, if the statements involve the word “All” and “Some”.

For example, “some horses are not brown”. The negation of the given statement is “All horses are brown”.

Practice Problems

Write the negation for the following sentences:

  1. Line a is parallel to line b.
  2. Some prime numbers are odd.
  3. 3+3 = 6.
  4. All irrational numbers are real numbers.
  5. No student is lazy.

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