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 $0< x<1$.
Write down the converse of following statements.
(i) If a rectangle ' $R$ ' is a square, then $R$ is a rhombus.
(ii) If today is Monday, then tomorrow is Tuesday.
(iii) If you go to Agra, then you must visit Taj Mahal.
(iv) If sum of squares of two sides of a triangle is equal to the square of third side of a triangle, then the triangle is right angled.
(v) If all three angles of a triangle are equal, then the triangle is equilateral.
(vi) If $x: y=3: 2$, then $2 x=3 y$.
(vii) If $S$ is a cyclic quadrilateral, then the opposite angles of $S$ are supplementary.
(viii) If $x$ is zero, then $x$ is neither positive nor negative.
(ix) If two triangles are similar, then the ratio of their corresponding sides are equal.
(i) If thes rectangle ' $R$ ' is rhombus, then it is square.
(ii) If tomorrow is Tuesday, then today is Monday.
(iii) If you must visit Taj Mahal, you go to Agra.
(iv) If the triangle is right angle, then sum of squares of two sides of a triangle is equal to the square of third side.
(v) If the triangle is equilateral, then all three angles of triangle are equal.
(vi) If $2 x=3 y$, then $x: y=3: 2$
(vii) If the opposite angles of a quadrilateral are supplementary, then $S$ is cyclic.
(viii) If $x$ is neither positive nor negative, then $x$ is 0 .
(ix) If the ratio of corresponding sides of two triangles are equal, then triangles are similar.