M.C.Q (1 Marks)

MCQ 511 Mark

Let $A=\{1,2,3,4\}$ and $R$ be a relation in $A$ given by $R=\{(1,1),(2,2),(3,3),(4,4),(1,2)$, $(2,1),(3,1)\}$. Then, $R$ is

✓
Reflexive only
B
Symmetric only
C
An equivalence relation
D
None of these

Answer

Correct option: A.

Reflexive only

(a) : Reflexive: $(1,1),(2,2),(3,3),(4,4) \in R$;
$\therefore \quad R$ is reflexive.

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MCQ 521 Mark

Let $L$ denote the set of all straight lines in a plane. Let a relation $R$ be defined by $\alpha R \beta \Leftrightarrow \alpha \perp \beta, \alpha, \beta \in L$. Then, $R$ is

A
Reflexive only
✓
Symmetric only
C
Transitive only
D
None of these

Answer

Correct option: B.

Symmetric only

(b) : Given, $\alpha R \beta \Leftrightarrow \alpha \perp \beta \Leftrightarrow \beta \perp \alpha \Rightarrow \beta R \alpha$ Hence, $R$ is symmetric.

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MCQ 531 Mark

The maximum number of equivalence relations on the set $A=\{1,2,3\}$ are

A
1
B
2
C
3
✓
5

Answer

Correct option: D.

(d): The smallest equivalence relation is the identity relation $R_1=\{(1,1),(2,2),(3,3)\}$
Then, two ordered pairs of two distinct elements can be added to give three more equivalence relations.
$
R_2=\{(1,1),(2,2),(3,3),(1,2),(2,1)\}
$
Similarly $R_3$ and $R_4$.
Finally the largest equivalence relation, that is the universal relation.
$
\begin{array}{l}
R_5=\{(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(3,1),(2,3), \\
(3,2)\}
\end{array}
$

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MCQ 541 Mark

Let $T$ be the set of all triangles in the Euclidean plane, and let a relation $R$ on $T$ be defined as $a R b$ if $a$ is congruent to $b \forall a, b \in T$. Then $R$ is

A
reflexive but not transitive
B
transitive but not symmetric
✓
an equivalence relation
D
None of these

Answer

Correct option: C.

an equivalence relation

(c) : (i) We know that every triangle is congruent to itself.
$\therefore \quad\left(T_1, T_1\right) \in R$ for all $T_1 \in T$. Thus, $R$ is reflexive.
(ii) Let $\left(T_1, T_2\right) \in R$
$\Rightarrow \quad T_1$ is congruent to $T_2$.
$\Rightarrow \quad T_2$ is congruent to $T_1$.
$\therefore \quad\left(T_2, T_1\right) \in R$
Thus, $R$ is symmetric.
(iii) Let $\left(T_1, T_2\right) \in R$ and $\left(T_2, T_3\right) \in R$.
$\Rightarrow \quad T_1$ is congruent to $T_2$ and $T_2$ is congruent to $T_3$.
$\therefore \quad T_1$ is congruent to $T_3$
$\Rightarrow \quad\left(T_1, T_3\right) \in R$.
Thus, $R$ is transitive.
$\therefore \quad R$ is an equivalence relation.

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MCQ 551 Mark

For the set $A=\{1,2,3\}$, define a relation $R$ on the set $A$ as follows :
$R=\{(1,1),(2,2),(3,3),(1,3)\}$
How many ordered pairs to be added to $R$ to make it the smallest equivalence relation?

✓
1
B
2
C
3
D
4

Answer

Correct option: A.

(a): Here, $A=\{1,2,3\}$ and the relation $R=\{(1,1),(2,2)(3,3)(1,3)\}$.
Clearly, $R$ is reflexive but not symmetric as $(1,3) \in R$ but $(3,1) \notin R$.
We shall include $(3,1)$ to the above relation to make it smallest equivalence relation
$
R^{\prime}=\{(1,1),(2,2),(3,3)(1,3),(3,1)\} \text {. }
$
$R^{\prime}$ is certainly transitive as transitivity is not contradicted.

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Maths M.C.Q (1 Marks)