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Maths Solve the Following Question.(2 Marks)

Relations · Maharashtra Board STD 12 (MSBSHSE - Semi English Medium). 36 questions.

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Solve the Following Question.(2 Marks)

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36 questions · self-marked practice — reveal the answer and mark yourself.

Question 12 Marks
Write the smallest equivalence relation on the set A = {1, 2, 3}.
Answer
The smallest equivalence relation on the set A = {1, 2, 3} is R = {(1, 1), (2, 2), (3, 3)}.
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Question 22 Marks
Define a symmetric relation.
Answer
A relation R on a set A is said to be symmetric if $\text{a, b}\in\text{R}$
Implies that, $\text{b, a}\in\text{R}$ for all $\text{a, b}\in\text{A}$
That is, aRb implies that bRa for all $\text{a, b}\in\text{A}$
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Question 32 Marks
Let $R = {(x, y): |x^2 - y^2| < 1}$ be a relation on set $A = {1, 2, 3, 4, 5}$. Write $R$ as a set of ordered pairs.
Answer
Given: $R = {(x, y): |x^2 - y^2| < 1}$ be a relation on $A = {1, 2, 3, 4, 5}$
Then, $R = {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5)}$
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Question 42 Marks
Write the domain of the relation R defined on the set Z of integers as follows:
$(a, b) \in R ⇔ a^2 + b^2 = 25$
Answer
We have,
$R=\left\{(a, b) \in R \Leftrightarrow a^2+b^2=25\right\}$ be a relation on $Z$.
The domain of $R$ is the value of ' $a$ ' $\in Z$, that satisfies $a^2+b^2=25$
$a^2+b^2=25$
$\Rightarrow a= \pm \sqrt{25-b^2}$
$\therefore \text { Domain of } R=\{0, \pm 3, \pm 4, \pm 5\}$
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Question 52 Marks
If R = {(x, y): x + 2y = 8} is a relation on N by, then write the range of R.
Answer
R = {(x, y): x + 2y = 8, $\text{x, y}\in\text{N}$}
Then, the values of y can be 1, 2, 3 only.
Also, y = 4 cannot result in x = 0 because x is a natural number.
Therefore, range of R is {1, 2, 3}.
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Question 62 Marks
State the reason for the relation R on the set {1, 2, 3} given by R = {(1, 2), (2, 1)} to be transitive.
Answer
Since $1,2\in\text{R,}$ $2,1\in\text{R}$ but $1,1\notin\text{R,}$ R is not transitive on the set 1, 2, 3. For R to be in a transitive relation, there must have $1,1\in\text{R.}$
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Question 72 Marks
If A = {1, 2, 3, 4} define relations on A which have properties of being:
Symmetric but neither reflexive nor transitive.
Answer
The relation on A having properties of being symmetric, but neither reflexive nor transitive is,
R = {(1, 2), (2, 1)}
The relation R on A is neither reflexive nor transitive, but symmetric.
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Question 82 Marks
The following defines a relation on N:
xy is square of an integer, $\text{x, y}\in\text{N}$
Determine which of the above relations are reflexive, symmetric and transitive.
Answer
A relation R in A is said to be reflexive if aRa for all $\text{a}\in\text{A},$ R is symmetric if aRb ⇒ bRa, for all $\text{a, b}\in\text{A}$ and it is said to be transitive if aRb and bRc ⇒ aRc for all $\text{a, b, c}\in\text{A.}$xy is square of an integer, $\text{x, y}\in\text{N}$
$(\text{x, y})\in\big\{(1, 1), (2, 2), (4, 1), (1, 4), (3, 3), (9, 1), (1, 9), (4, 4), (2, 8), \$8, 2), (16, 1), (1, 16), .....\big\}$ $$
This is reflexive as (1, 1), (2, 2), .... are present.
This is also symmetric because if aRb ⇒ bRa, for all $\text{a, b}\in\text{N.}$
This is transitive also because if aRb and bRc ⇒ aRc for all $\text{a, b, c}\in\text{N.}$
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Question 92 Marks
Let the relation R be defined on N by aRb if 2a + 3b = 30. Then write R as a set of ordered pairs.
Answer
If $\text{a, b}\in\text{N}$ then b must be an even integer so that $\text{a}\in\text{N}$
Hence only possible values for b are 2, 4, 6, 8.
if b = 2, it gives a = 12
if b = 4, it gives a = 9
if b = 6, it gives a = 6
if b = 8, it gives a = 3
Hence $(\text{a, b})\in\big\{(3, 8), (6, 6), (9, 4), (12, 2)\big\}$ $$
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Question 102 Marks
Let $R=\left\{\left(a, a^3\right)\right.$ : $a$ is a prime number less than 5$\}$ be a relation. Find the range of $R$.
Answer
We have,
$R=\left\{\left(a, a^3\right): a\right.$ is a prime number less than 5$\}$
Or,
$R=\{(2,8),(3,27)\}$
So, the range of $R$ is $\{8,27\}$.
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Question 112 Marks
Let the relation $R$ be defined on the set $A=\{1,2,3,4,5\}$ by $R=\left\{(a, b):\left|a^2-b^2\right|<8\right\}$. Write $R$ as a set of ordered pairs.
Answer
Given: $A=\{1,2,3,4,5\} R=\left\{(a, b):\left|a^2-b^2\right|<8\right\} R=\{(1,1),(1,2),(2,1),(2,2),(2,3),(3,2),(3,3),(3,4),(4,3),(4,4),(5, 5)\}$
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Question 122 Marks
Write the smallest reflexive relation on set $A=\{1,2,3,4\}$.
Answer
The smallest reflexive relation $R$ on any set $A$ is the identity relation $I_A$ on the set $A$.
We are given, $A=\{1,2,3,4\}$
$\therefore R=\{(1,1),(2,2),(3,3),(4,4)\}$
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Question 132 Marks
Define an equivalence relation.
Answer
A relation R on a set A is said to be equivalence relation on a if R is:
Reflexive, Symmetric and Transitive.
R = {(x, y): x = y} on the set of real numbers is an equivalence relation.
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Question 142 Marks
The following defines a relation on N:
$\text{x} +\text{y} = 10,\text{x, y}\in\text{N}$ $$
Determine which of the above relations are reflexive, symmetric and transitive.
Answer
A relation R in A is said to be reflexive if aRa for all $\text{a}\in\text{A},$ R is symmetric if aRb ⇒ bRa, for all $\text{a, b}\in\text{A}$ and it is said to be transitive if aRb and bRc ⇒ aRc for all $\text{a, b, c}\in\text{A.}$$\text{x} +\text{y} = 10,\text{x, y}\in\text{N}$
$(\text{x, y})\in\big\{(1, 9), (9, 1), (2, 8), (8, 2), (3, 7), (7, 3),\\ (4, 6), (6, 4), (5, 5)\big\}$ $$
This is not reflexive as (1, 1), (2, 2), .... are absent.
This only follows the condition of symmetric set as $(1,9)\in\text{R}$ also $(9,1)\in\text{R}$ Similarly other cases are also satisfy the condition.
This is not transitive because {(1, 9), (9, 1)} $\in\text{R}$ but (1, 1) is absent.
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Question 152 Marks
If A = {3, 5, 7} and B = {2, 4, 9} and R is a relation given by "is less than", write R as a set ordered pairs.
Answer
Since, R = x, y: x, y $\in\text{N}$ and x < y,
Hence, R = {(3, 4), (3, 9), (5, 9), (7, 9)}
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Question 162 Marks
Give an example of a relation which is,Reflexive and transitive but not symmetric.
Answer
Let R be the relation on A such that
R = {(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 3)}
Clearly, the relation R on A is reflexive and transitive, but not symmetric.
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Question 172 Marks
Let A = {a, b, c} and the relation R be defined on A as follows: R = {(a, a), (b, c), (a, b)}. Then, write minimum number of ordered pairs to be added in R to make it reflexive and transitive.
Answer
We have,
A = {a, b, c} and R = {(a, a), (b, c), (a, b)}.
R can be a reflexive relation only when elements (b, b) and (c, c) are added to it.
R can be a transitive relation only when the element (a, c) is added to it.
So, the minimum number of ordered pairs to be added in R is 3.
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Question 182 Marks
Define a reflexive relation.
Answer
A relation R on a set A is said to be reflexive if every element of A is related to itself.
Mathematically, reflexive relation is written as R = {(a, a): for all $\text{a}\in\text{A}$}
For example if A = {1, 2, 3}, then a reflexive relation on A will be R = {(1, 1), (2, 2), (3, 3)}
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Question 192 Marks
If A = {1, 2, 3, 4} define relations on A which have properties of being:
Reflexive, symmetric and transitive.
Answer
The relation on A having properties of being symmetric, reflexive and transitive is,
R = {(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1)}
The relation R is an equivalence relation on A.
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Question 202 Marks
Let R be the equivalence relation on the set Z of the integers given by R = {(a, b): 2 divides a - b}. Write the equivalence class [0].
Answer
$\text{a, b}\in\text{Z}$ and R is given by R = {(a, b): 2 divides a - b}.The equivalence classes can be taken as [0], [1].
Note that, $\text{for}\ 0\leq\text{i}\leq1,$ [i] = {2n + i: $\text{n}\in\text{Z}$}
So equivalence class [0] = {2n: $\text{n}\in\text{Z}$}
It is clear that all the elements of equivalence class [0] are even.
Hence, equivalence class $[0]=\{0,\pm2,\pm4,\pm6\ ...\}$
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Question 212 Marks
Give an example of a relation which is,
Transitive but neither reflexive nor symmetric.
Answer
Let R be the relation on A such that
R = {(1, 2), (2, 3), (1, 3)}
The relation R on A is transitive, but neither symmetric nor reflexive.
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Question 222 Marks
The following defines a relation on N:
$\text{x}>\text{y, x, y}\in\text{N}$
Determine which of the above relations are reflexive, symmetric and transitive.
Answer
A relation R in A is said to be reflexive if aRa for all $\text{a}\in\text{A},$ R is symmetric if aRb ⇒ bRa, for all $\text{a, b}\in\text{A}$ and it is said to be transitive if aRb and bRc ⇒ aRc for all $\text{a, b, c}\in\text{A.}$$\text{x}>\text{y, x, y}\in\text{N}$
$(\text{x, y})\in\big\{(2, 1), (3, 1),..., (3, 2), (4, 2),....\big\}$ $$
This is not reflexive as (1, 1), (2, 2), .... are absent.
This is not symmetric as (2, 1) is present but (1, 2) is absent.
This is transitive as $(3,2)\in\text{R}$ and $(2,1)\in\text{R}$ also $(3,1)\in\text{R},$ similarly this property satisfies all cases.
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Question 232 Marks
If $A=\{2,3,4\}, B=\{1,3,7\}$ and $R=\{(x, y): x \in A, y \in B$ and $x<y\}$ is a relation from $A$ to $B$, then write $R^{-1}$.
Answer
Since $R=\{(x, y): x \in A, y \in B$ and $x<y\}$
$R=\{(2,3),(2,7),(3,7),(4,7)\}$
Hence, $R^{-1}=\{(3,2),(7,2),(7,3),(7,4)\}$
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Question 242 Marks
Let A = {2, 3, 4, 5} and B = {1, 3, 4}. If R is the relation from A to B given by a R b if "a is a divisor of b". Write R as a set of ordered pairs.
Answer
We have, A = {2, 3, 4, 5}, B = {1, 3, 4} and relation from A to B is given by aRb if ''is divisor of'' B$\therefore$ R can be written as ordered pair as R = {(2, 4), (3, 3), (4, 4)}
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Question 252 Marks
Let $A=\{3,5,7\}, B=\{2,6,10\}$ and $R$ be a relation from $A$ to $B$ defined by $R=\{(x, y)$ : $x$ and $y$ are relatively prime $\}$. Then, write $R$ and $R^{-1}$.
Answer
$R=\{(x, y): x$ and $y$ are relatively prime $\}$
Then,
$R=\{(3,2),(5,2),(7,2),(3,10),(7,10),(5,6),(7,6)\}$
So, $R^{-1}=\{(2,3),(2,5),(2,7),(10,3),(10,7),(6,5),(6,7)\}$
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Question 262 Marks
Let A = {0, 1, 2, 3} and R be a relation on A defined as R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R reflexive? symmetric? transitive?
Answer
We have, R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}As, $(\text{a, a})\in\text{R}\ \forall\ \text{a}\in\text{A}$
So, R is a reflexive relation. Also, $(\text{a, b})\in\text{R}$ and $(\text{b, a})\in\text{R}$ So, R is a symmetric as well And, $(0,1)\in\text{R}$ but $(1,2)\notin\text{R}$ and $(2,3)\notin\text{R}$ So, R is not a transitive relation.
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Question 272 Marks
The following defines a relation on N:
x + 4y = 10, $\text{x, y}\in\text{N}$
Determine which of the above relations are reflexive, symmetric and transitive.
Answer
A relation R in A is said to be reflexive if aRa for all $\text{a}\in\text{A},$ R is symmetric if aRb ⇒ bRa, for all $\text{a, b}\in\text{A}$ and it is said to be transitive if aRb and bRc ⇒ aRc for all $\text{a, b, c}\in\text{A.}$$\text{x} + 4\text{y} = 10, \ \text{x, y}\in\text{N}$, $$
$(\text{x, y})\in\big\{(6, 1), (2, 2)\big\}$ $$
This is not reflexive as (1, 1), (2, 2), .... are absent.
This is also symmetric because $(6,1)\in\text{R}$ but (1, 6) is absent.
This is not transitive as there are only two elements in the set having no element common.
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Question 282 Marks
Define a transitive relation.
Answer
A relation R on a set A is said to be transitive if
$(\text{a, b})\in\text{R}$ and $(\text{b, c})\in\text{R}$
$\Rightarrow\ (\text{a, c})\in\text{R}$ for all $\text{a, b, c}\in\text{R}$
i.e., aRb and bRc
⇒ aRc for all $\text{a, b, c}\in\text{R}$
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Question 292 Marks
If $R=\left\{(x, y): x^2+y^2 \leq 4 ; x, y \in Z\right\}$ is a relation on $Z$, write the domain of $R$.
Answer
Domain of R is the set of values of x that satisfies the relation R.
Because x must be an integer, the provided values of x are:
$0,\pm1,\pm2$
Thus, Domain of R is $0,\pm1,\pm2$
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Question 302 Marks
Write the identity relation on set A = {a, b, c}.
Answer
Identity set of A is:
I = {(a, a), (b, b), (c, c)}
Every element of this relation is related to itself.
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Question 312 Marks
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)}
Write the ordered pairs to be added to R to make the smallest equivalence relation.
Answer
(3, 1) is the single ordered pair which needs to be added to R to make it the smallest equivalence relation.
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Question 322 Marks
Let A = {1, 2, 3} and R = {(1, 2), (1, 1), (2, 3)} be a relation on A. What minimum number of ordered pairs may be added to R so that it may become a transitive relation on A.
Answer
We have, A = {1, 2, 3} and R = {(1, 2), (1, 1), (2, 3)}
To make R transitive we shall add (1, 3) only.
$$ $\therefore \text{R}' = \big\{(1, 2), (1, 1), (2, 3), (1, 3)\big\}$
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Question 332 Marks
Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons.
Answer
No, it is not necessary that a relation which is symmetric and transitive is reflexive as well.
For Example,
Let $A=\{a, b, c\}$ be a set and
$R_2=\{(a, a)\}$ is a relation defined on $A$.
Clearly,
$R _2$ is symmetric and transitive but not reflexive.
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Question 342 Marks
Give an example of a relation which is,
Symmetric and transitive but not reflexive.
Answer
Let R be the relation on A such that,
R = {(1, 2), (2, 1), (1, 3), (3, 1), (2, 3)}
We see that the relation R on A is symmetric and transitive, but not reflexive.
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Question 352 Marks
If $R$ is a symmetric relation on a set A, then write a relation between $R$ and $R^{-1}$.
Answer
Here, $R$ is symmetric on the set A.
Let $(\text{a, b})\in\text{R}$
$\Rightarrow\ (\text{b, a})\in\text{R}$ [Since $R$ is symmetric]
$\Rightarrow\ (\text{a, b})\in\text{R}^{-1}$ [By definition of inverse relation]
$\Rightarrow\ \text{R}\subset\text{R}^{-1}$
Let $(\text{x, y})\in\text{R}^{-1}$
$\Rightarrow\ (\text{y, x})\in\text{R}$ [By definition of inverse relation]
$\Rightarrow\ (\text{x, y})\in\text{R}$ [Since $R$ is symmetric]
$\Rightarrow\ \text{R}^{-1}\subset\text{R}$
Thus, $\text{R}=\text{R}^{-1}$
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Question 362 Marks
A = {1, 2, 3, 4, 5, 6, 7, 8} and if R = {(x, y): y is one half of x; x, y ∈ A} is a relation on A, then write R as a set of ordered pairs.
Answer
Since R = {(x, y): y is one half of x; x, y ∈ A}
So, R = {(2, 1), (4, 2), (6, 3), (8, 4)}
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