Question

Check whether the following sentence is a statement? Justify

(i) A triangle has three sides

(ii) $0$ is a complex number

(iii) Sky is red

(iv) Every set is an infinite set

(v) $15+8>23$

(vi) $y+9=7$

(vii) Where is your bag?

(viii) Every square is a rectangle

(ix) Sum of opposite angles of a cyclic quadrilateral is ${180}^{\circ }$

(x) ${\sin }^{2}x+{\cos }^{2}x=0$

Answer 1

(i)

Using the definition of statement which is "A statement is a sentence which is either true or false but not both simultaneously".

Given sentence: "A triangle has three sides"

Given sentence is true.
Hence given sentence ""A triangle has three sides" represent a statement.


(ii)

Using the definition of statement which is "A statement is a sentence which is either true or false but not both simultaneously.
Given sentence: "0 is a complex number ".
$0=0+i\left(0\right)$
Here $0$ is in the form of $a+ib$

So,given sentence is true.
Hence given sentence "$0$ is a complex number" represents a statement.


(iii)

Using the definition of statement which is "A statement is a sentence which is either true or false but not both simultaneously".

Given sentence: "Sky is red".
Given sentence is false.

Hence given sentence "Sky is red" represents a statement.

(iv)

Using the definition of statement which is "A statement is a sentence which is either true or false but not both simultaneously".
Given sentence: "Every set is an infinite set".

Given sentence is false as set can be finite also.
Hence given sentence "Every set is an infinite set" represents a statement.

(v)

Using the definition of statement which is "A statement is a sentence which is either true or false but not both simultaneously".

Given sentence:$"15+8>23"$
Clearly $15+8=23$

So ,given sentence is false.
Hence given sentence $"15+8>23"$ represents a statement.


(vi)

Using the definition of statement which is "A statement is a sentence which is either true or false but not both simultaneously".

Given sentence: $"y+9=7"$
$y+9=7$ cannot be assigned as true or false without knowing the value of $y$.

Hence given sentence $"y+9=7"$ cannot be called a statement.


(vii)

Using the definition of statement which is "A statement is a sentence which is either true or false but not both simultaneously".
Given sentence: "Where is your bag?"
"Where is your bag?"cannot be assigned as true or false (In fact, it is a question and Interrogative sentence is not considered as statement)

Hence given sentence "Where is your bag?"cannot be called a statement.


(viii)

Using the definition of statement which is "A statement is a sentence which is either true or false but not both simultaneously".

Given sentence: "Every Square is a rectangle"

Given sentence is true.
Hence given sentence "Every Square is a rectangle" represents a statement.
.
(ix)


Using the definition of statement which is "A statement is a sentence which is either true or false but not both simultaneously".

Given sentence: "Sum of opposites sides of a cyclic quadrilateral is ${180}^{\circ }$".

Given sentence is always true.

Hence given sentence "Sum of opposites sides of a cyclic quadrilateral is ${180}^{\circ }$" represents a statement.


(x)

Using the definition of statement which is "A statement is a sentence which is either true or false but not both simultaneously".

Given sentence: ${}^{\prime \prime }{\sin }^{2}x+{\cos }^{2}x=0"$

We know that, ${\sin }^{2}x+{\cos }^{2}x=1$ (Trigonometric identity)

So, given sentence is always false.
Hence given sentence $"{\sin }^{2}x+{\cos }^{2}x=0"$
represents a statement.