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.
Write down the negation of following compound statements.
(i) All rational numbers are real and complex.
(ii) All real numbers are rationals or irrationals.
(iii) $x=2$ and $x=3$ are roots of the quadratic equation $x^2-5 x+6=0$.
(iv) A triangle has either 3 -sides or 4 -sides.
(v) 35 is a prime number or a composite number.
(vi) All prime integers are either even or odd.
(vii) $|x|$ is equal to either $x$ or $-x$.
(viii) 6 is divisible by 2 and 3 .
(i) Let $p$ : All rational numbers are real.
$q$ : All rational numbers are complex.
$\sim p$ : All rational number are not real.
$\sim q$ : All rational numbers are not complex.
$\sim(p \wedge q):$ All rational numbers are not real or not complex. $\quad[\because \sim(p \wedge q)=\sim p \vee \sim q]$
(ii) Let $p$ : All real numbers are rationals.
$q$ : All real numbers are irrational.
Then, the negation of the above statement is given by
$\sim(p \vee q)$ : All real numbers are not rational and all real numbers are not irrational.
$$[\because \sim(p \vee q)=\sim p \wedge \sim q]$$
(iii) Let $p$ : $x=2$ is root of quadratic equation $x^2-5 x+6=0$.
$q: x=3$ is root of quadratic equation $x^2-5 x+6=0$.
Then, the negation of conjunction of above statement is given by
$\sim(p \wedge q): x=2$ is not a root of quadratic equation $x^2-5 x+6=0$ or $x=3$ is not a root of the quadratic equation $x^2-5 x+6=0$.
(iv) Let $p$ : A triangle has 3 -sides.
$q$ : A triangle has 4 -sides.
Then, negation of disjunction of the above statement is given by $\sim(p \vee q)$ : A triangle has neither 3-sides nor 4-sides.
(v) Let $p: 35$ is a prime number.
$q: 35$ is a composite number.
Then, negation of disjunction of the above statement is given by $\sim(p \vee q): 35$ is not a prime number and it is not a composite number.
(vi) Let $p$ : All prime integers are even.
$q$ : All prime integers are odd.
Then negation of disjunction of the above statement is given by
$\sim(p \vee q)$ : All prime integers are not even and all prime integers are not odd.
(vii) Let $p:|x|$ is equal to $x$.
$q:|x|$ is equal to $-x$.
Then negation of disjunction of the above statement is given by
$\sim(p \vee q):|x|$ is not equal to $x$ and it is not equal to $-x$.
(viii) Let $p: 6$ is divisible by 2 .
$q: 6$ is divisible by 3 .
Then, negation of conjunction of above statement is given by
$\sim(p \wedge q): 6$ is not divisible by 2 or it is not divisible by 3
Rewrite each of the following statements in the form of conditional statements.
(i) The square of an odd number is odd.
(ii) You will get a sweet dish after the dinner.
(iii) You will fail, if you will not study.
(iv) The unit digit of an integer is 0 or 5 , if it is divisible by 5 .
(v) The square of a prime number is not prime.
(vi) $2 b=a+c$, if $a, b$ and $c$ are in AP.
We know that, some of the common expressions of conditional statement $p \rightarrow q$ are
(i) if $p$, then $q$
(ii) $q$ if $p$
(iii) ponly if $q$
(iv) $p$ is sufficient for $q$
(v) $q$ is necesary for $p$
(vi) $\sim q$ implies $\sim p$
So, use above information to get the answer
(i) If the number is odd number, then its square is odd number.
(ii) If you take the dinner, then you will get sweet dish.
(iii) If you will not study, then you will fail.
(iv) If an integer is divisible by 5 , then its unit digits are 0 or 5 .
(v) If the number is prime, then its square is not prime.
(vi) If $a, b$ and $c$ are in $A P$, then $2 b=a+c$.
Form the biconditional statement $p \leftrightarrow q$, where
(i) $p$ : The unit digits of an integer is zero.
$q:$ It is divisible by 5 .
(ii) $p$ : A natural number $n$ is odd.
$q$ : Natural number $n$ is not divisible by 2 .
(iii) $p$ : A triangle is an equilateral triangle.
$q$ : All three sides of a triangle are equal.
(i) $p \leftrightarrow q$ : The unit digit of on integer is zero, if and only if it is divisible by 5 .
(ii) $p \leftrightarrow q$ : A natural number no odd if and only if it is not divisible by 2 .
(iii) $p \leftrightarrow q$ : A triangle is an equilateral triangle if and only if all three sides of triangle are equal.
Write down the contrapositive of the following statements.
(i) If $x=y$ and $y=3$, then $x=3$.
(ii) If $n$ is a natural number, then $n$ is an integer.
(iii) If all three sides of a triangle are equal, then the triangle is equilateral.
(iv) If $x$ and $y$ are negative integers, then $x y$ is positive.
(v) If natural number $n$ is divisible by 6 , then $n$ is divisible by 2 and 3 .
(vi) If it snows, then the weather will be cold.
(vii) If $x$ is a real number such that $0< x<1$, then $x^2<1$.
(i) If $x \neq 3$, then $x \neq y$ or $y \neq 3$.
(ii) If $n$ is not an integer, then it is not a natural number.
(iii) If the triangle is not equilateral, then all three sides of the triangle are not equal.
(iv) If $x y$ is not positive integer, then either $x$ or $y$ is not negative integer.
(v) If natural number $n$ is not divisible by 2 or 3 , then $n$ is not divisible by 6 .
(vi) The weather will not be cold, if it does not snow.
(vii) If $x^2 \nleq 1$, then $x$ is not a real number such that $0< x<1$.