Rewrite the following statements in the form "p if and only if q". p: If you watch television, then your mind is free and if your mind is free, then you watch television.
Answer
You watch television if and only if your mind is free.
Show that the following statement is true by the method of contrapositive p: "If x is an integer and $x^2$ is odd, then x is also odd"
Answer
Let q and r be the statements given by q : If x is an integer and $x ^2$ is odd $r : x$ is an odd integer. Then, p : "If q , thenr." If possible, let $r$ be false. Then, $r$ is false $\Rightarrow x$ is not an odd integer $\Rightarrow x$ is an even integer $\Rightarrow x=(2 n)$ for some integer $n$ $\Rightarrow x^2=4 n^2 \Rightarrow x^2$ is an even integer $\Rightarrow q$ is false. Thus, $r$ is false $\Rightarrow q$ is false. Hence, $p$ : "if $q$, then $r$ " is a true statement.
Show that the statement: p : "If x is a real number such that $x ^3+ x =0$, then x is $0^{\text {" }}$ is true by. Method of contradition.
Answer
Let q and r be the statements given q : x is a real number such that $x ^3+ x =0 . r : x$ is 0 . Then, p : if q , then r . Metnod of contradiction: If possible, let $p$ be not true. Then, $p$ is not true $\Rightarrow-$ pis true $\Rightarrow-(p \Rightarrow r)$ is true $\Rightarrow q$ and $-r$ is true $\Rightarrow x$ is a real number such that $x ^3+ x =0$ and $x =0 \Rightarrow x =0$ and $x \neq 0$ This a contradiction. Hence, p is true.
Which of the following statements are true and which are false? In each case give a valid reason for saying so: s: If x and y are integers such that x > y, then -x < -y.
Answer
True. Because, for any two integers, if x - y is positive then -(x - y) is negative.
Check the validity of the following statements: r: 60 is a multiple of 3 or 5.
Answer
The statement is: r: 60 ism ultiple of 3 or 5 is a com pound statement of the following statements: p: 60 is multiple of 3 q: 60 is multiple of 5 Suppose q is false. That is, 60 is not a multiple of 5. Clearly p is true. Thus, if we assume that q is false, then p is true. Hence, the statement is true i.e. the statement "r" is a valid statement.
For the following statements, determine whether an inclusive "OR" or exclusive "OR" is used. Give reasons for your answer. Students can take Hindi or Sanskrit as their third language.
Answer
Exclusive OR is used because students can opt for either Hindi or Sanskrit as their third language.
Which of the following statements are true and which are false? In each case give a valid reason for saying so: r: Circle is a particular case of an ellipse.
Answer
True. Because a circle is an ellipse that has equal axes.
Write the component statements of the following compound statements and check whether the compound statement is true or false: $x = 2$ and $x = 3$ are the roots or the equation $3x^2 − x − 10 = 0$.
Answer
The component statements of the given compound statement are:
$x = 2$ is the root or the equation $3x^2 - x - 10 = 0$.
$x = 3$ is the root or the equation $3x^2 - x - 10 = 0$.
The connective used is "and". So, both component statements must be true for the compound statement to be true. The statement "$x = 3x = 3$ is the root or the equation $3x^2 - x - 10 = 0$" is false. Therefore, the compound statement is false.
For the following statements, determine whether an inclusive "OR" or exclusive "OR" is used. Give reasons for your answer. To apply for a driving licence, you should have a ration card or a passport.
Answer
Inclusive OR because a person could have both ration card as well as passport.
Show that the statement: $p$ : "If $x$ is a real number such that $x^3+x=0$, then $x$ is 0 " is true by. Method of contrapositive
Answer
Let q and r be the statements given $\mathrm{q}: \mathrm{x}$ is a real number such that $\mathrm{x}^3+\mathrm{x}=0 . \mathrm{r}: \mathrm{x}$ is 0 . Then, p : if q , then r . Method of contrapositive: Let $r$ be not true. Then, $r$ is not true. $\Rightarrow x \neq 0, x \in R \Rightarrow x\left(x^2+1\right) \neq 0, x \in R \Rightarrow q$ is not true Thus, $-r=-q$. Hence, $p: q \Rightarrow r$ is true.
Write the component statements of the following compound statements and check whether the compound statement is true or false: To enter into a public library children need an identity card from the school or a letter from the school authorities.
Answer
The component statements of the given compound statement are:
To enter into a public library, children need an identity card from the school.
To enter into a public library, children need a letter from the school authorities.
The compound statement is true because both component statements are true.
Rewrite the following statements in the form "p if and only if q". q: If a quadrilateral is equiangular, then it is a rectangle and if a quadrilateral is a rectangle, then it is equiangular.
Answer
A quadrilateral is a rectangle if and only if it is equiangular.
Check the validity of the following statements: q: 125 is a multiple of 5 and 7.
Answer
The statement is: "125 is multiple of 5 and 7" Since 125 is a multiple of 5 but it is not a multiple of 7. So, q is not a true statement i.e. the statement "q" is not a valid statement.
Which of the following statements are true and which are false? In each case give a valid reason for saying so: q: The centre of a circle bisects each chord of the circle.
Answer
False. Because, a chord does not have to pass through the centre.
State the converse and contrapositive of the following statements: If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Answer
Converse: If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. Contrapositive: If the diagonals of a quadrilateral do not bisect each other, then it is not a parallelogram.
Write the component statements of the following compound statements and check whether the compound statement is true or false: Square of an integer is positive or negative.
Answer
The component statements of the given compound statement are:
Square of an integer is positive.
Square of an integer is negative.
The compound statement is true because the first statement is true. Since the connective used is "or" and one of the component statements is true, the compound statement is true.
State the converse and contrapositive of the following statements: A positive integer is prime only if it has no divisors other than 1 and itself.
Answer
Converse: If a positive integer has no divisors other than 1 and itself, then it is prime. Contrapositive: If a positive integer has some divisors other than 1 and itself, then it is not prime.