Exercise 12.3-Additional Problems - Chapter 12 Discrete Mathematics 12th Maths Guide Samacheer Kalvi Solutions - SaraNextGen [2024-2025]

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Additional Problems
Choose the correct or the most suitable answer from the given four alternatives.
Question 1.
Which of the following are statements?
(i) May God bless you
(ii) Rose is a flower
(iii) milk is white
(iv) 1 is a prime number
(a) (i), (ii), (iii)
(b) (i), (ii), (iv)
(c) (i), (iii), (iv)
(d) (ii), (iii), (iv)

Solution:
(d) (ii), (iii), (iv)
Hint:
Sentence (ii), (iii) and (iv) are statements
(ii) Rose is a flower - True
(iii) Milk is white - True
(iv) 1 is a prime number
$\therefore$ (ii), (iii), (iv) are statements
(i) May god bless you. This statement can not be assigned True or False.
$\therefore$ (i) is not a statements

Question 2.
If a compound statement is made up of the three simple statements, then the number of rows in the truth table is
(a) 8
(b) 6
(c) 4
(d) 2

Solution:
(a) 8
Hint:
The number of rows in truth table $=2^{\mathrm{n}}=2^3=8$
Question 3.
If $\mathrm{p}$ is $\mathrm{T}$ and $\mathrm{q}$ is $\mathrm{F}$, then which of the following have the truth value $T$ ?

(1) $p \vee q$
(2) $\sim p \vee q$
(3) $p \vee \sim q$
(4) $p \wedge \sim q$
(a) (i), (ii), (iii)
(b) (i), (ii), (iv)
(c) (i), (iii), (iv)
(d) (ii), (iii), (iv)

Solution:
(c) (i), (iii), (iv)
Hint:
$\mathrm{p}$ is $\mathrm{T}$ then $\sim \mathrm{p}$ is $\mathrm{F}$
$\mathrm{q}$ is $\mathrm{F}$ then $\sim \mathrm{q}$ is $\mathrm{T}$
$p \vee q$ is $\mathrm{T}$
$-p \vee q$ is $\mathrm{F}$
$p \vee \sim q$ is $\mathrm{T}$
$p \wedge \sim q$ is $\mathrm{T}$
Question 4.
The number of rows in the truth offimg 6 is
(a) 2
(b) 4
(c) 6
(d) 8

Solution:
(b) 4
Hint:
Number of simple statements given is 2. i.e., $p$ and $q$.
Number of rows in the truth table of $\sim[p \wedge(\sim q)]=2^2=4$
Question 5.
The conditional statement $\mathrm{p} \rightarrow \mathrm{q}$ is equivalent to
(1) $p \vee q$
(2) $p \vee \sim q$
(3) $\sim p \vee q$
(4) $p \wedge q$

Hint: Truth table for $p \rightarrow q$
(a) Truth table for $p \vee q$

(b) Truth table for $p \vee \sim q$

(c) Truth table for $\sim p \vee q$

(d) Truth table for $p \wedge q$

The truth table for $\mathrm{p} \rightarrow \mathrm{q}$ and $(\sim p \vee q)$ having the last column identical. $\therefore \mathrm{p} \rightarrow \mathrm{q}$ is equivalent to $(\sim p \vee q)$
Solution:
(3) $\sim p \vee q$
Question 6.
Which of the following is a tautology?
(a) $p \vee q$
(b) $p \wedge q$
(c) $p \vee \sim p$
(d) $p \wedge \sim p$
Solution:
(c) $\boldsymbol{p} \vee \sim \boldsymbol{p}$
Hint:
A statement is said to be a tautology if the last column of its truth table contains only $T$.

(1) If $p$ is $\mathrm{F}$ and $q$ is $\mathrm{F}$ then $p \vee q$ is $\mathrm{F}$
$\therefore p \vee q$ is not a tautology.
(2) If $p$ is $\mathrm{F}$ and $q$ is $\mathrm{F}$ then $p \wedge q$ is $\mathrm{F}$
$\therefore p \wedge q$ is not a tautology.
(3) If $p$ is $\mathrm{T}$ then $\sim p$ is $\mathrm{F}$ then $p \vee \sim p$ is $\mathrm{T}$.
If $p$ is $\mathrm{F}$ then $\sim p$ is $\mathrm{T}$ then $p \vee \sim p$ is $\mathrm{T}$.
$\therefore p \vee \sim q$ is a tautology.
(4) If $p$ is $\mathrm{T}$ then $\sim p$ is $\mathrm{F}$ then $p \wedge \sim p$ is $\mathrm{F}$.
If $p$ is $\mathrm{F}$ then $\sim p$ is $\mathrm{T}$ then $p \wedge \sim p$ is $\mathrm{F}$.
$\therefore p \wedge \sim p$ is not a tautology.
Question 7.
In the set of integers with operation * defined by $a * b=a+b-a b$, the value of $3 *(4 * 5)$ is .......
(a) 25
(b) 15
(c) 10
(d) 5
Hint:

Solution:
(a) 25
$
\begin{aligned}
& \mathrm{a} * \mathrm{~b}=\mathrm{a}+\mathrm{b}-\mathrm{ab} \\
& 3 *(4 * 5)=3 *(4+5-4(5)) \\
& =3 *(9-20) \\
& =3 *(-11) \\
& =3+(-11)-3(-11) \\
& =3-11+33 \\
& =-8+33=25
\end{aligned}
$
Question 8.
In the multiplicative group of cube root of unity, the order of $\omega^2$ is ......
(a) 4
(b) 3
(c) 2
(d) 1
Solution:
(b) 3
Hint:
In the cube root of unity $\omega^3=1$
$
1+\omega+\omega^2=0
$

$\begin{aligned}
\left(\omega^2\right)^2 & =\omega^4=\omega^3 \cdot \omega=\omega \\
\left(\omega^2\right)^3 & =\omega^6=\left(\omega^3\right)^2=(1)^2=1 \\
\therefore 0\left(\omega^2\right) & =3
\end{aligned}$

Question 10.
In the set of real numbers $\mathrm{R}$, an operation * is defined by $a^* b=\sqrt{a^2+b^2}$. Then the value of $(3 * 4) * 5$ is
(a) 5
(b) $5 \sqrt{2}$
(c) 25
(d) 50

Solution:
(b) $5 \sqrt{2}$
Hint:
$
\begin{aligned}
a * b & =\sqrt{a^2+b^2} \\
(3 * 4) * 5 & =\left(\sqrt{3^2 * 4^2}\right) * 5=5 * 5=\sqrt{5^2+5^2}=\sqrt{25+25}=\sqrt{50}=5 \sqrt{2}
\end{aligned}
$
Question 11.
The order of $-1$ in the multiplicative group of $4^{\text {th }}$ roots of unity is
(a) 4
(b) 3
(c) 2
(d) 1

Solution:
(a) 4
Hint:
The roots of fourth roots of unity are $1,-1, i,-1$
The identity element is 1
$(-i)^4=i^4=1$
Order of $(-i)=4$.