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

Updated By SaraNextGen

On April 24, 2024, 11:35 AM

Ex $12.3$
Choose the correct or the most suitable answer from the given four alternatives.
Question 1.
A binary operation on a set $\mathrm{S}$ is a function from ......
(a) $\mathrm{S} \rightarrow \mathrm{S}$
(b) $(\mathrm{S} \times \mathrm{S}) \rightarrow \mathrm{S}$
(c) $\mathrm{S} \rightarrow(\mathrm{S} \times \mathrm{S})$
(d) $(\mathrm{S} \times \mathrm{S}) \rightarrow(\mathrm{S} \times \mathrm{S})$
Solution:
(b) $(\mathrm{S} \times \mathrm{S}) \rightarrow \mathrm{S}$
Question 2.
Subtraction is not a binary operation in
(a) $R$
(b) $\mathrm{Z}$
(c) $\mathrm{N}$
(d) $Q$
Solution:
(c) $\mathrm{N}$
Hint:
For example $2,5 \in N$ but $2-5=3 \notin N$
Question 3.
Which one of the following is a binary operation on $\mathrm{N}$ ?
(a) Subtraction
(b) Multiplication
(c) Division
(c) All the above
Solution:
(b) Multiplication
Question 4.
In the set $\mathrm{R}$ of real numbers ' ${ }^{\prime}$ ' is defined as follows. Which one of the following is not a binary operation on R?
(a) $\mathrm{a}^* \mathrm{~b}=\min (\mathrm{a} \cdot \mathrm{b})$
(b) $\mathrm{a} * \mathrm{~b}=\max (\mathrm{a}, \mathrm{b})$
(c) $\mathrm{a} * \mathrm{~b}=\mathrm{a}$

(d) $\mathrm{a}^* \mathrm{~b}=\mathrm{a}^{\mathrm{b}}$
Solution:
(d) $a * b=a^b$
Hint:
Since $-2,1 / 2 \in R$, but $(-2)^{1 / 2} \notin R$
Question 5.
The operation $*$ defined by $\mathrm{a} * \mathrm{~b}=\frac{a b}{7}$ is not a binary operation on
(a) $\mathrm{Q}^{+}$
(b) $\mathrm{Z}$
(c) $R$
(c) $\mathrm{C}$
Solution:
(b) $\mathrm{Z}$
Hint:
Since $3,5 \in Z$, but $\frac{3 \times 5}{7} \notin Z$.
Question 6.
In the set $Q$ define $a \odot b=a+b+a b$. For what value of $y, 3 \odot(y \odot 5)=7$ ?
(a) $y=\frac{2}{3}$
(b) $y=\frac{-2}{3}$
(c) $y=\frac{-3}{2}$
(d) $y=4$
Hint: $\quad a \odot b=a+b+a b$
So, $\quad y \odot 5=y+5+5 y$
Now, $3 \odot(y \odot 5)=3 \odot x=3+x+3 x$
$=3+(5+6 y)+3(5+6 y)$
$=3+5+6 y+15+18 y=7$ (Given)
$\Rightarrow \quad 24 y=7-23=-16$
$\Rightarrow \quad y=\frac{-16}{24}=\frac{-2}{3}$
Solution:
(b) $y=\frac{-2}{3}$

Question 7.
If $\mathrm{a} * \mathrm{~b}=\sqrt{a^2+b^2}$ on the real numbers then $*$ is
(a) commutative but not associative
(b) associative but not commutative
(c) both commutative and associative
(d) neither commutative nor associative
Let, a, b $\in \mathrm{R}$
$\begin{array}{ll}\text { Now } & a * b=\sqrt{a^2+b^2} \\ \Rightarrow & b * a=\sqrt{b^2+a^2} \\ \Rightarrow & a * b=b * a \quad \Rightarrow * \text { is commutative }\end{array}$
Now to find $a *(b * c)$
Now $\quad b * c=\sqrt{b^2+c^2}=\mathrm{D}$ (Say)
So $\quad a *(b * c)=a * \mathrm{D}=\sqrt{a^2+\mathrm{D}^2}$
$
=\sqrt{a^2+b^2+c^2}
$
To find $(a * b) * c$
$
(a * b)=\sqrt{a^2+b^2}=\mathrm{E} \text { (Say) }
$
Now $\quad(a * b) * c=\mathrm{E} * c=\sqrt{\mathrm{E}^2+c^2}$
$
=\sqrt{a^2+b^2+c^2}
$
$(1)=(2)=*$ is associative
So * is both commutative and associative
Solution:
(c) both commutative and associative

Question 8.
Which one of the following statements has the truth value $T$ ?
(a) $\sin \mathrm{x}$ is an even function.
(b) Every square matrix is non-singular
(c) The product of complex number and its conjugate is purely imaginary
(d) $\sqrt{5}$ is an irrational number
Solution:
(d) $\sqrt{5}$ is an irrational number
Question 9.
Which one of the following statements has truth value $F$ ?
(a) Chennai is in India or $\sqrt{2}$ is an integer
(b) Chennai is in India or $\sqrt{2}$ is an irrational number
(c) Chennai is in China or $\sqrt{2}$ is an integer
(d) Chennai is in China or $\sqrt{2}$ is an irrational number
Solution:
(c) Chennai is in China or $\sqrt{2}$ is an integer
Question 10.
If a compound statement involves 3 simple statements, then the number of rows in the truth table is
(a) 9
(b) 8
(c) 6
(d) 3
Solution:
(b) 8
Hint:
$
\text { (i.e.) } 2^3=8
$

Question 11.
Which one is the inverse of the statement $(p \vee q) \rightarrow(p \wedge q)$ ?
(a) $(p \wedge q) \rightarrow(p \vee q)$
(b) $\neg(p \vee q) \rightarrow(p \wedge q)$
(c) $(\neg p \vee \neg q) \rightarrow(\neg p \wedge \neg q)$
(d) $(\neg p \wedge \neg q) \rightarrow(\neg p \vee \neg q)$
Solution:
(a) $(\neg p \wedge \neg q) \rightarrow(\neg p \vee \neg q)$
Question 12 .
Which one is the contrapositive of the statement $(p \vee q) \rightarrow r$ ?
(a) $\neg \boldsymbol{r} \rightarrow(\neg p \wedge \neg q)$
(b) $\neg r \rightarrow(\neg p \vee q)$
(c) $r \rightarrow(p \wedge q)$
(d) $p \rightarrow(q \vee r)$
Solution:
(a) $\neg r \rightarrow(\neg p \wedge \neg q)$

Question 13.
The truth table for $(p \wedge q) \vee \neg q$ is given below

Which one of the following is true?
(a) (b) (c) (d)
(1) $\mathrm{T} \quad \mathrm{T} \quad \mathrm{T} \quad \mathrm{T}$
(2) $T \quad F \quad T \quad T$
(3) $T \quad T \quad F \quad T$
(4) $T, F \quad F \quad F$
Hint: The truth table for $(p \wedge q) \vee \neg q$

Solution:
(3) T T F T
Question 14.
In the last column of the truth table for $\neg(p \vee \neg q)$ the number of final outcomes of the truth value ' $F$ ' are (a) 1
(b) 2
(c) 3
(d) 4
Hint:
The truth table for $\neg(p \vee \neg q)$

Solution:
(c) 3
Question 15.
Which one of the following is incorrect? For any two propositions $\mathrm{p}$ and $\mathrm{q}$, we have
(a) $\neg(p \vee q) \equiv \neg p \wedge \neg q$
(b) $\neg(p \wedge q) \equiv \neg p \vee \neg q$
(c) $\neg(p \vee q) \equiv \neg p \vee \neg q$
(d) $\neg(\neg p) \equiv p$
Solution:
(c) $\neg(p \vee q) \equiv \neg p \vee \neg q$
Question 16.

Which of the following is correct for the truth $(p \wedge q) \rightarrow \neg p$ ?
, (a) (b) (c) (d)
(1) $\mathrm{T} \quad \mathrm{T} \quad \mathrm{T} \quad \mathrm{T}$
(2) $\mathbf{F} \quad \mathbf{T} \quad \mathbf{T} \quad \mathbf{T}$
(3) $\mathrm{F} \quad \mathrm{F} \quad \mathrm{T} \quad \mathrm{T}$
(4) $\mathrm{T} T \mathrm{~T} \quad \mathrm{~F}$
Hint:

$\text { The truth table for }(p \wedge q) \rightarrow \neg p$

Solution:
(2) F T T T
Question 17.
The dual of $\neg(p \vee q) \vee[p \vee(p \wedge \neg r)]$ is
$(a) \neg(p \wedge q) \wedge[p \vee(p \wedge \neg r)]$
(b) $(p \wedge q) \wedge[p \wedge(p \vee \neg r)]$
(c) $\neg(p \wedge q) \wedge[p \wedge(p \wedge r)]$
(d) $\neg(p \wedge q) \wedge[p \wedge(p \vee \neg r)]$
Solution:
(d) $\neg(p \wedge q) \wedge \mid p \wedge(p \vee \neg r)]$
Question 18.
The proposition $p \wedge(\neg p \vee q)]$ is .......
(a) a tautology
(b) a contradiction
(c) logically equivalent to $p \wedge q$
(d) logically equivalent to $p \vee q$
Solution:
(c) logically equivalent to $p \wedge q$
Question 19.
Determine the truth value of each of the following statements:
(a) $4+2=5$ and $6+3=9$
(b) $3+2=5$ and $6+1=7$
(c) $4+5=9$ and $1+2=4$
(d) $3+2=5$ and $4+7=11$

Solution:
(1) F T F T

Question 20 .
Which one of the following is not true?
(a) Negation of a negation of a statement is the statement itself.
(b) If the last column of the truth table contains only $T$ then it is a tautology.
(c) If the last column of its truth table contains only $F$ then it is a contradiction
(d) If $\mathrm{p}$ and $\mathrm{q}$ are any two statements then $\mathrm{p} \leftrightarrow \mathrm{q}$ is a tautology.
Solution:
(d) If p and q are any two statements then $\mathrm{p} \leftrightarrow \mathrm{q}$ is a tautology.