Question 1: Find out which of the following sentences are statements and which are not. Justify your answer.

(i) Listen to me, Ravi!           (ii) Every set is a finite set.

(iii) Two non-empty sets have always a non-empty intersection.

(iv) The pussy cat is black.      (v) Are all circles round?

(vi) Alt triangles have three sides.            (vii) Every rhombus is a square.

(viii) \displaystyle x^2 +5 |x| + 6 = 0 has no real roots.           (ix) This sentence is a statement.

(x) Is the earth round?            (xi) Go!

(xii) The real number \displaystyle x is less than 2.            (xiii) There are 35 days in a month.

(xiv) Mathematics is difficult.            (xv) All real numbers are complex numbers.

(xvi) The product of (-1) and 8 is 8.

Answer:

(i) Listen to me, Ravi!

The sentence “Listen to me, Ravi! “Is an exclamatory sentence. Therefore it is not a statement

(ii) Every set is a finite set.

This sentence is always false, because there are sets which are not finite. Therefore it is a statement.

(iii) Two non-empty sets have always a non-empty intersection.

This sentence is always false, because there are non-empty sets whose intersection is empty. Therefore it is a statement.

(iv) The pussy cat is black.

There are some cats which are black, and not black, So, the given sentence may or may not be true. Therefore it is not a statement.

(v) Are all circles round?

The given sentence is an interrogative sentence. Therefore it is not a statement.

(vi) All triangles have three sides.

The given sentence is a true declarative sentence. Therefore it is a statement.

(vii) Every rhombus is a square.

This sentence is always false, because Rhombuses are not a square. Therefore it is a statement.

(viii) \displaystyle x^2 +5 |x| + 6 = 0 has no real roots.

If you solve the equation, you will see there are no real roots. Therefore it is a statement.

(ix) This sentence is a statement.

The statement is not indicating the correct value, hence we can say that value contradicts the sense of the sentence. Therefore it is not a statement.

(x) Is the earth round?

The given sentence is an interrogative sentence. Therefore it is not a statement.

(xi) Go!

It is an exclamatory sentence. Therefore it is not a statement.

(xii) The real number \displaystyle x is less than 2.

We cannot determine whether the sentence is true or false without knowing the value of x. Therefore it is not a statement.

(xiii) There are 35 days in a month.

It is a false assertive sentence. Therefore it is a false statement.

(xiv) Mathematics is difficult.

Mathematics could be difficult for some and easy for some people.  So the sentence cannot be determined as a true or false sentence. Therefore it is not a statement.

(xv) All real numbers are complex numbers.

It is true as we can write a real number as x+i0. Therefore it is a true statement.

(xvi) The product of (-1) and 8 is 8.

The sentence is always false as (-1) times 8 is (-8). Therefore it is a false statement.

\\

Question 2: Give three examples of sentences which are not statements. Give reasons for the answers.

Answer:

The three examples of sentences which are not statements are as follows:

i) May God bless you! – This sentence is an optive. So, we cannot assign a true or false to it and hence it is not a statement.

ii) Three plus four is six – The sentence is a false declarative sentence. Hence it is not a statement.

iii) Give me a glass of water – This sentence is an imperative sentence. It either expresses a request or a command. Hence it is not a statement.

(iv) How beautiful! – This is an exclamatory sentence. Hence not a statement.

(v) Open the door. – This is an order. Hence not a statement.

(vi) Where are you going? – This is an interrogative sentence. Hence not a statement.