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.
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.
(i) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
The sentence " Listen to me, Ravi !' is an excalamatory sentence. So, it is not a statement.
(ii) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
The sentence is always false, because there are sets which are not finite. Hence, it is a statement.
(iii) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
This sentence is always false, because there are non- empty sets whose intersection is empty.
Hence, it is a statement.
(iv) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
Some cats are black and some not. So, the given sentence may or may not be true. Hence, it is a statement.
(v) A statement or a proposition is an assertive (or a declarative) sentence which iseither true or false but not both.
The sentence " Are all circle round ? " is an interrogative sentence. So, it not a statement.
(vi) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
The "All triangles have three sides." is a true declarative sentence. So, it is a true ststement.
(vii) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
This sentence is always false, because there are rhombuses that are not squares. Hence, it is a statement.
(viii) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
If x > 0,
x2+5|x|+6=0
⇒x2+5x+6=0
⇒x=−3orx=−2
But, since x > 0, the equation has no roots.
If x <0,
x2+5|x|+6=0
⇒x2+5x+6=0
which has no real roots.
So, the sentence x2−5x+6=0 is always true.
Hence, it is a statement.
(ix) It is not'a statement.
The sentence "This sentence is a statement." Cannot be assigned a truth value of either true or false, because either assignment contradicts the sense of the sentence.
(x) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
The sentence "Is the earth round? " is an interrogative sentence. So, it is not a statement.
(xi) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
The sentence "Go !" is an exclamatory sentences. So, it is not a statement.
(xii) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
"The real number x is less than 2" is not a statement, because its truth or falsity cannot be confirmed without knowing the value of x.
(xiii) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
"There are 35 days in a month" is a false declarative sentence. So, it is a false statement.
(xiv) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
"Mathematics is difficult" is true for those who may not like mathematics. But, for others, it may not be true. So, the given sentence may or may not be true. Hence, it is not a statement.
(xv) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
This sentence is always true. Hence, it is a statement.
(xvi) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
This sentence is always false, because (−1×8=−8). Hence, it is a statement.
(i) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
The sentence " Listen to me, Ravi !' is an excalamatory sentence. So, it is not a statement.
(ii) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
The sentence is always false, because there are sets which are not finite. Hence, it is a statement.
(iii) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
This sentence is always false, because there are non- empty sets whose intersection is empty.
Hence, it is a statement.
(iv) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
Some cats are black and some not. So, the given sentence may or may not be true. Hence, it is a statement.
(v) A statement or a proposition is an assertive (or a declarative) sentence which iseither true or false but not both.
The sentence " Are all circle round ? " is an interrogative sentence. So, it not a statement.
(vi) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
The "All triangles have three sides." is a true declarative sentence. So, it is a true ststement.
(vii) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
This sentence is always false, because there are rhombuses that are not squares. Hence, it is a statement.
(viii) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
If x > 0,
x2+5|x|+6=0
⇒x2+5x+6=0
⇒x=−3orx=−2
But, since x > 0, the equation has no roots.
If x <0,
x2+5|x|+6=0
⇒x2+5x+6=0
which has no real roots.
So, the sentence x2−5x+6=0 is always true.
Hence, it is a statement.
(ix) It is not'a statement.
The sentence "This sentence is a statement." Cannot be assigned a truth value of either true or false, because either assignment contradicts the sense of the sentence.
(x) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
The sentence "Is the earth round? " is an interrogative sentence. So, it is not a statement.
(xi) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
The sentence "Go !" is an exclamatory sentences. So, it is not a statement.
(xii) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
"The real number x is less than 2" is not a statement, because its truth or falsity cannot be confirmed without knowing the value of x.
(xiii) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
"There are 35 days in a month" is a false declarative sentence. So, it is a false statement.
(xiv) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
"Mathematics is difficult" is true for those who may not like mathematics. But, for others, it may not be true. So, the given sentence may or may not be true. Hence, it is not a statement.
(xv) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
This sentence is always true. Hence, it is a statement.
(xvi) A statement or a proposition is an assertive (or a declarative) sentence which is either true or false but not both.
This sentence is always false, because (−1×8=−8). Hence, it is a statement.
