Question

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 cat pussy is black.
(v) Are all circles round?
(vi) All triangles have three sides.
(vii) Every rhombus is a square.
(viii) x² + 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 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 1

(i) Listen to me, Ravi!
It is an exclamatory sentence. Therefore, it is not a statement.

(ii) Every set is a finite set.
It is a false assertive sentence because there are some sets that are infinite like the set of all real numbers. Therefore, it is a statement.

(iii) Two non-empty sets have always a non-empty intersection.
It is a false assertive sentence. Two non-empty sets with no common elements can have an empty intersection. Therefore, it is a statement.

(iv) The pussy cat is black.
It is a declarative sentence, which may be true or false but cannot be both at the same time, so it is a statement.

(v) Are all circles round?
It is an interrogative sentence, so it is not a statement.

(vi) All triangles have three sides.
It is a true declarative sentence because a figure that has three sides is a triangle. Thus, it is a true statement.

(vii) Every rhombus is a square.
It is not true that every rhombus is a square because some rhombi may have all angles other than 90. So, it is a false statement.

(viii) x² + 5 | x | + 6 = 0 has no real roots.
It is a true declarative sentence, so it is a statement.

(ix) This sentence is a statement.
Without knowing the sentence, we cannot decide whether it is true or false. So, it is not a statement.

(x) Is the earth round?
It is an interrogative sentence, so it is not a statement.

(xi) Go!
It is an exclamatory sentence, so it is not a statement.

(xii) The real number x is less than 2.
We cannot decide whether this sentence is true or false without knowing the value of $x$. So, it is not a statement.

(xiii) There are 35 days in a month.
It is a false assertive sentence, so it is a false statement.

(xiv) Mathematics is difficult.
Mathematics could be easy for some people, so this sentence may or may not be true. So, it is not a statement.

(xv) All real numbers are complex numbers.
It is true because we can write a real number as $x+0i$. So, it is a true statement.

(xvi) The product of (−1) and 8 is 8.
It is an assertive sentence; therefore, it is a statement. But $-1\times 8=-8$; therefore, the statement is false.