\displaystyle \textbf{Question 1: }\text{Find out which of the following sentences are statements and which are not.}
\displaystyle \text{Justify your answer.}
\displaystyle \text{(i) Listen to me, Ravi!}\qquad\text{(ii) Every set is a finite set.}
\displaystyle \text{(iii) Two non-empty sets always have a non-empty intersection.}
\displaystyle \text{(iv) The pussy cat is black.}\qquad\text{(v) Are all circles round?}
\displaystyle \text{(vi) All triangles have three sides.}\qquad\text{(vii) Every rhombus is a square.}
\displaystyle \text{(viii) }x^2+5|x|+6=0\text{ has no real roots.}
\displaystyle \text{(ix) This sentence is a statement.}\qquad\text{(x) Is the earth round?}
\displaystyle \text{(xi) Go!}\qquad\text{(xii) The real number }x\text{ is less than }2.
\displaystyle \text{(xiii) There are }35\text{ days in a month.}\qquad\text{(xiv) Mathematics is difficult.}
\displaystyle \text{(xv) All real numbers are complex numbers.}
\displaystyle \text{(xvi) The product of }(-1)\text{ and }8\text{ is }8.
\displaystyle \text{Answer:}

\displaystyle \text{(i) ``Listen to me, Ravi!'' is an imperative sentence expressing a command.}
\displaystyle \therefore \text{It cannot be assigned a truth value and is not a statement.}

\displaystyle \text{(ii) There are infinite sets, such as the set of natural numbers.}
\displaystyle \therefore \text{The sentence is false, but it is a statement.}

\displaystyle \text{(iii) Two non-empty sets can be disjoint and hence can have an empty intersection.}
\displaystyle \text{For example, }\{1\}\cap\{2\}=\varnothing.
\displaystyle \therefore \text{The sentence is false, but it is a statement.}

\displaystyle \text{(iv) The sentence makes an assertion about a particular cat.}
\displaystyle \text{It must be either true or false, even if its truth value is not known to us.}
\displaystyle \therefore \text{It is a statement.}

\displaystyle \text{(v) ``Are all circles round?'' is an interrogative sentence.}
\displaystyle \therefore \text{It cannot be assigned a truth value and is not a statement.}

\displaystyle \text{(vi) Every triangle has three sides by definition.}
\displaystyle \therefore \text{It is a true statement.}

\displaystyle \text{(vii) A rhombus need not have four right angles and hence need not be a square.}
\displaystyle \therefore \text{The sentence is false, but it is a statement.}

\displaystyle \text{(viii) Let }y=|x|,\text{ where }y\geq0.
\displaystyle x^2+5|x|+6=y^2+5y+6=(y+2)(y+3).
\displaystyle \text{Since }y\geq0,\text{ both }y+2\text{ and }y+3\text{ are positive.}
\displaystyle \therefore x^2+5|x|+6\neq0\text{ for every real }x.
\displaystyle \therefore \text{The equation has no real roots, so it is a true statement.}

\displaystyle \text{(ix) ``This sentence is a statement'' is a declarative sentence.}
\displaystyle \text{It asserts that it is a statement, and this assertion is true.}
\displaystyle \therefore \text{It is a true statement.}

\displaystyle \text{(x) ``Is the earth round?'' is an interrogative sentence.}
\displaystyle \therefore \text{It cannot be assigned a truth value and is not a statement.}

\displaystyle \text{(xi) ``Go!'' is an imperative sentence expressing a command.}
\displaystyle \therefore \text{It cannot be assigned a truth value and is not a statement.}

\displaystyle \text{(xii) The truth value of }x<2\text{ depends on the value assigned to }x.
\displaystyle \therefore \text{It is an open sentence and is not a statement.}

\displaystyle \text{(xiii) No month has }35\text{ days.}
\displaystyle \therefore \text{The sentence is false, but it is a statement.}

\displaystyle \text{(xiv) Mathematics may be difficult for some people and easy for others.}
\displaystyle \therefore \text{The sentence has no definite truth value and is not a statement.}

\displaystyle \text{(xv) Every real number }x\text{ can be written as }x+0i.
\displaystyle \therefore \text{Every real number is a complex number, so it is a true statement.}

\displaystyle \text{(xvi) }(-1)\times8=-8\neq8.
\displaystyle \therefore \text{The sentence is false, but it is a statement.}
\displaystyle \\

\displaystyle \textbf{Question 2: }\text{Give three examples of sentences which are not statements.}
\displaystyle \text{Give reasons for your answers.}
\displaystyle \text{Answer:}
\displaystyle \text{Examples of sentences which are not statements are:}

\displaystyle \text{(i) May God bless you!}
\displaystyle \text{This is an optative sentence. It cannot be assigned a truth value.}
\displaystyle \therefore \text{It is not a statement.}

\displaystyle \text{(ii) Give me a glass of water.}
\displaystyle \text{This is an imperative sentence expressing a request.}
\displaystyle \therefore \text{It is not a statement.}

\displaystyle \text{(iii) Where are you going?}
\displaystyle \text{This is an interrogative sentence asking a question.}
\displaystyle \therefore \text{It is not a statement.}
\displaystyle \text{Additional examples:}

\displaystyle \text{(iv) How beautiful!}
\displaystyle \text{This is an exclamatory sentence expressing a feeling.}
\displaystyle \therefore \text{It is not a statement.}

\displaystyle \text{(v) Open the door.}
\displaystyle \text{This is an imperative sentence expressing a command.}
\displaystyle \therefore \text{It is not a statement.}

\displaystyle \text{(vi) Go!}
\displaystyle \text{This is an imperative sentence expressing a command.}
\displaystyle \therefore \text{It is not a statement.}
\displaystyle \\

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