Which of the following sentences are statements? 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) $\sin ^2 x+\cos ^2 x=0$
As we know, a statement is a sentence which is either true or false but not both simultaneously.
(i) It is true statement.
(ii) It is true statement.
(iii) It is false statement.
(iv) It is false statement.
(v) It is false statement.
(vi) $y+9=7$ It is not considered as a statement, since the value of $y$ is not given.
(vii) It is a question, so it is not a statement.
(viii) It is a true statement.
(ix) It is true statement.
$(x)$ It is false statement.
Find the component statements of the following compound statements.
(i) Number 7 is prime and odd.
(ii) Chennai is in India and is the capital of Tamil Nadu.
(iii) The number 100 is divisible by 3,11 and 5.
(iv) Chandigarh is the capital of Haryana and UP.
(v) $\sqrt{7}$ is a rational number or an irrational number.
(vi) 0 is less than every positive integer and every negative integer.
(vii) Plants use sunlight, water and carbon dioxide for photosynthesis.
(viii) Two lines in a plane either intersect at one point or they are parallel.
(ix) A rectangle is a quadrilateral or a 5 sided polygon.
(i) $p$ : Number 7 is prime.
$q$ : Number 7 is odd.
(ii) P : Chennai is in India.
$q$ : Chennai is capital of Tamil Nadu.
(iii) $p: 100$ is divisible by 3 .
$q$ : 100 is divisible by 11 .
$r: 100$ is divisible by 5 .
(iv) $p$ : Chandigarh is capital of Haryana.
$q$ : Chandigarh is capital of UP.
(v) $p: \sqrt{7}$ is a rational number.
$q: \sqrt{7}$ is an irrational number.
(vi) $p: 0$ is less than every positive integer.
$q: 0$ is less than every negative integer.
(vii) $p$ : Plants use sunlight for photosysthesis.
$q$ : Plants use water for photosynthesis.
$r$ : Plants use carbon dioxide for photosysthesis.
(viii) $p$ : Two lines in a plane intersect at one point.
$q$ : Two lines in a plane are parallel.
(ix) $p$ : A rectangli, is a quadrilateral.
$q$ : A rectangle is a 5 -sided polygon.
Write the component statements of the following compound statements and check whether the compound statement is true or false.
(i) 57 is divisible by 2 or 3.
(ii) 24 is a multiple of 4 and 6 .
(iii) All living things have two eyes and two legs.
(iv) 2 is an even number and a prime number.
(i) Given compound statement is of the form ' $p v q$ '. Since, the statement ' $p v q$ ' has the truth value $T$ whenever either $p$ or $q$ or both have the truth value $T$.
So, it is true statement.
Its component statements are
$p: 57$ is divisible by 2 . [false]
$q: 57$ is divisible by 3 . [true]
(ii) Given compound statement is of the form ' $p \wedge q$ '. Since, the statement ' $p \wedge q$ ' have the truth value $T$ whenever both $p$ and $q$ have the truth value $T$. So, it is a true statement.
Its component statements are
$p: 24$ is multiple of 4 [true]
$q: 24$ is multiple of 6 . [true]
(iii) It is a false statement. Since ' $p \wedge q$ ' has truth value $F$ whenever either $p$ or $q$ or both have the truth value $F$.
Its component statements are
$p$ : All living things have two eyes. [false]
$q$ : All living things have two legs. [false]
(iv) It is a true statement.
Its component statements are
$p: 2$ is an even number. [true]
$q: 2$ is a prime number. [true]
Write the negative on the following simple statements.
(i) The number 17 is prime.
(ii) $2+7=6$.
(iii) Violets are blue.
(iv) $\sqrt{5}$ is a rational number.
(v) 2 is not a prime number.
(vi) Every real number is an irrational number.
(vii) Cow has four legs.
(viii) A leap year has 366 days.
(ix) All similar triangles are congruent.
(x) Area of a circle is same as the perimeter of the circle.
(i) The number 17 is not prime.
(ii) $2+7 \neq 6$.
(iii) Violets are not blue.
(iv) $\sqrt{5}$ is not a rational number.
(v) 2 is a prime number.
(vi) Every real number is not an irrational number.
(vii) Cow has not four legs.
(viii) A leap year has not 366 days.
(ix) There exist similar triangles which are not congruent.
(x) Area of a circle is not same as the perimeter of the circle.
Translate the following statements into symbolic form
(i) Rahul passed in Hindi and English.
(ii) $x$ and $y$ are even integers.
(iii) 2, 3 and 6 are factors of 12 .
(iv) Either $x$ or $x+1$ is an odd integer.
(v) A number is either divisible by 2 or 3 .
(vi) Either $x=2$ or $x=3$ is a root of $3 x^2-x-10=0$.
(vii) Students can take Hindi or English as an optional paper.
(i) $p$ : Rahul passed in Hindi.
$q$ : Rahul passed in English.
$p \wedge q$ : Rahul passed in Hindi and English.
(ii) $p: x$ is even integers.
$q: y$ is even integers.
$p \cap q: x$ and $y$ are even integers.
(iii) $p: 2$ is factor of 12 .
$q: 3$ is factor of 12 .
$r: 6$ is factor of 12 .
$p \wedge q \wedge r: 2,3$ and 6 are factor of 12 .
(iv) $p: x$ is an odd integer.
$q:(x+1)$ is an odd integer.
$p \vee q$ : Either $x$ or $(x+1)$ is an odd integer.
(v) $p$ : A number is divisible by 2 .
$q$ : A number is divisible by 3.
$p \vee q$ : A number is either divisible by 2 or 3 .
(vi) $p: x=2$ is a root of $3 x^2-x-10=0$.
$q: x=3$ is a root of $3 x^2-x-10=0$.
$p \vee q$ : Either $x=2$ or $x=3$ is a root of $3 x^2-x-10=0$.
(vii) $p$ : Students can take Hindi as an optional paper.
$q$ : Students can take English as an optional subject.
$p \vee q$ : Students can take Hindi or English as an optional paper.