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Special Words Or Phrases

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The connecting words which are found in compound statements like “And”, “Or”, etc. are often used in Mathematical Statements. These are called connectives. 

1)  The word “And”:
Let us look at a compound statement with “And”.
p: A point occupies a position and its location can be determined. 
The statement can be broken into two component statements as 
q : A point occupies a position.
r : Its location can be determined. 
Here, we observe that both statements are true.

2)  The word “Or”:
p: An ice cream or pepsi is available with a Thali in a restaurant.
This means that a person who does not want ice cream can have a pepsi along with Thali or one does not want pepsi can have an ice cream along with Thali. That is, who do not want a pepsi can have an ice cream. A person cannot have both ice cream and pepsi. This is called an exclusive “Or”.

Rule for the compound statement with ‘Or’:
For example, consider the following statement. 
p: Two lines intersect at a point or they are parallel 
The component statements are 
q: Two lines intersect at a point. 
r: Two lines are parallel. 
Then, when q is true r is false and when r is true q is false. 
Therefore, the compound statement p is true.

4)  Quantifiers :
Quantifiers are phrases like, “There exists” and “For all”. Another phrase which appears in mathematical statements is “there exists”. For example, consider the statement. p: There exists a rectangle whose all sides are equal. This means that there is atleast one rectangle whose all sides are equal. 
A word closely connected with “there exists” is “for every” (or for all). Consider a statement.

p: For every prime number p,`sqrt p` is an irrational number.

This means that if S denotes the set of all prime numbers, then for all the members p of the set S,`sqrt p` is an irrational number.

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