Question

All of us know that humans have evolved from the lower species over a very long period of time. The main asset that makes humans far more superior than the other species is the ability to reason. This ability of reasoning makes one person superior to the other and humans have a special power to express their views, emotions, thoughts, etc., by producing voice using a collection of words. 

I. Each collection of words, which makes sense, is a sentence.

Are all the sentences mathematically acceptable statements? 

If not, then which sentence is called a mathematically acceptable statement?

Some sentences are given below, which of the following sentences are statements?

(Group activity: 3, 4 students)

 (Discuss among group of three or four students to justify your answer) 

1. Come here and do this work.
2. Two plus five is nine.
3. Where is your bottle?
4. Paris is in America.
5. Sum of opposite angles of a cyclic quadrilateral is 180°.
6. 15 + y  > 20
7. 5 is a complex number.
8. x + 9 = 15
9. Every square is a rectangle.
10. Mathematics is fun.

II. For each of the following compound statements first identify the connecting words and then find component statements of the compound statement and check whether the compound statement is true or false. (Group activity: 3, 4 students)

 Discuss among groups to get the answer. 

1. 36 is a multiple of 6 and 7.
2. 36 is a multiple of 6 or 7.
3. The school is closed, if there is a holiday or Sunday.
4. The number 100 is divisible by 3, 11 and 25.
5. Plants use sunlight, water and oxygen for photosynthesis.
6. All living things have two eyes and two legs.
7. 0 is less than every positive integer or every negative integer.
8. Calcutta is in India or is the capital of Maharashtra.
9. 5 is an even number and a prime number.
10. 5 is an even number or a prime number.

III. Can you get negation of all these compound statements given above?

Discuss among the group and write negation of all these compound statements.

You must be aware that the connector “Or” is used in two ways,

1. Inclusive “Or” 
2. Exclusive “Or” 

IV. Identify in the following statements, whether the “Or” used is “exclusive “or” inclusive. 

Discuss among the group the reasons for your answer.

1. A rectangle is a quadrilateral or a 5-sided polygon.
2. √7 is a rational number or an irrational number.
3. All integers are positive or negative.
4. To enter into a public library children need an identity card from the school or a letter from the school Principal.
5. Two lines intersect at a point or are parallel.

V. Write the contrapositive and converse of the following statements. 

(Group activity - 3, 4 students)

 Discuss among the group to get the answer. 

1. Something that is cold implies that it has a low temperature.
2. If a triangle is equilateral, it is isosceles.
3. If a number is divisible by 15, then it is divisible by 5.
4. x is an even number implies that x is divisible by 4.
5. If y is a prime number, then y is odd.
6. If it is hot outside, then you feel thirsty.
7. A positive integer is prime only if it has no divisors other than 1 and itself.