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Statements - Lesson Summary

Statement: A sentence is called a mathematically acceptable statement if it is either true or false.

Consider the sentences:

2 is a factor of 6.

The sum of three angles of a triangle is two right angles.

All sides of a rectangle are equal.

The radius of a circle is twice its diameter.

Hyderabad is far from here.

In these sentences, the first two sentences are true, while third and fourth sentences are false.

In the last sentence, it is not certain to say how far Hyderabad is from another place, unless it is mentioned.
Therefore, the first four sentences are statements.

Hyderabad's distance is different from different places.

Thus, this sentence is not a statement.

Mathematical statements are denoted by p,q,r,s,...

p: Sum of two even natural numbers is an even natural number.

q: The Taj Mahal is in Agra.

Simple statement: A statement is called simple if it cannot be broken into two or more statements.

Consider the statements:

p: 2 is an even number.

q: 2 is not an even number.

Statement p is true, while statement q is the denying statement p. This implies that statement q is false.

Such a statement is called the negation of a statement.

Negation of a statement: The denial of a statement is called the negation of the statement.

If p is a statement, then the negation of p is also a statement, and is denoted by ~p. This is read as 'not p'.

If p is true, then ~p is false.

If p is false, then ~p is true.

Negations of some statements

p: 5 is an odd number.

Commonly, phrases like, "It is not the case" or "It is false that" are used while forming the negation of a statement.

~p: It is not the case that 5 is an odd number.


~p: It is false that 5 is an odd number.


~p: 5 is not an odd number.

Compound statement: A mathematical statements obtained by combining one or more simple statements using some connecting words like "and", "or", etc. is called a compound statement.

Consider the statements,

p: 5 is both an odd and a prime number.

q: 5 is an odd number.

r: 5 is a prime number.

The above statements are connected with the word "and".

The simple statements that form the compound statement are called component statements.

Combine the given statements using the word "or".

p: x is a rational number.

q: x is an irrational number.

The compound statement:

r: x is either a rational or an irrational number.


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