Relations - Maharashtra Board STD 12 Maths Questions

Q 1MCQ1 Mark

The relation $S$ defined on the set $R$ of all real number by the rule $aSb$ iff $a ≥ b$ is:

A
An equivalence relation.
✓
Reflexive, transitive but not symmetric.
C
Symmetric, transitive but not reflexive.
D
Neither transitive nor reflexive but symmetric.

Answer: B.

Q 2MCQ1 Mark

$S$ is a relation over the set $R$ of all real numbers and it is given by $(\text{a, b})\in\text{S}\Leftrightarrow\text{ab}\geq0.$ Then, $S$ is:

A
Symmetric and transitive only.
B
Reflexive and symmetric only.
C
Antisymmetric relation.
✓
An equivalence relation.

Answer: D.

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Q 3MCQ1 Mark

The relation $R = \{(1, 1), (2, 2), (3, 3)\}$ on the set $\{1, 2, 3\}$ is:

A
Symmetric only.
B
Reflexive only.
✓
An equivalence relation.
D
Transitive only.

Answer: C.

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Q 4MCQ1 Mark

Let $L$ denote the set of all straight lines in a plane. Let a relation $R$ be defined by $lRm$ if $l$ is perpendicular to $m$ for all $l, m \in L.$ Then, $R$ is:

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

Answer: B.

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Q 5MCQ1 Mark

In the set $Z$ of all integers, which of the following relation $R$ is not an equivalence relation?

✓
$xRy :$ if $\text{x}\leq\text{y}$
B
$xRy :$ if $x = y$
C
$xRy :$ if $x - y$ is an even integer
D
$xRy :$ if $\text{x}\equiv\text{y}\ (\text{mod 3})$

Answer: A.

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Q 6Solve the Following Question.(2 Marks)2 Marks

Write the smallest equivalence relation on the set A = {1, 2, 3}.

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Q 7Solve the Following Question.(2 Marks)2 Marks

Define a symmetric relation.

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Q 8Solve the Following Question.(2 Marks)2 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.

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Q 9Solve the Following Question.(2 Marks)2 Marks

Write the domain of the relation R defined on the set Z of integers as follows:
$(a, b) \in R ⇔ a^2 + b^2 = 25$

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Q 10Solve the Following Question.(2 Marks)2 Marks

If R = {(x, y): x + 2y = 8} is a relation on N by, then write the range of R.

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Q 11Solve the Following Question.(3 Marks)3 Marks

If A = {1, 2, 3, 4} define relations on A which have properties of being:
Reflexive, transitive but not symmetric.

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Q 12Solve the Following Question.(3 Marks)3 Marks

Test whether the following relations $R_2$ are:

Reflexive.
Symmetric.
Transitive.

$R_2$ on $Z $ defined by $(\text{a, b})\in\text{R}_2\Leftrightarrow\ |\text{a}-\text{b}|\leq5$

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Q 13Solve the Following Question.(3 Marks)3 Marks

Let $A = \{1, 2, 3\},$ and let $R_1 = \{(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)\}$. Find whether or not the relations $R_1$ on $A$ is:

Reflexive.
Symmetric.
Transitive.

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Q 14Solve the Following Question.(3 Marks)3 Marks

The following relation are defined on the set of real numbers.
aRb if $1 + ab > 0$
Find whether these relation are reflexive, symmetric or transitive.

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Q 15Solve the Following Question.(3 Marks)3 Marks

Let $A=\{1,2,3\}$, and let $R_3=\{(1,3),(3,3)\}$. Find whether or not the relations $R_3$ on $A$ is:
i. Reflexive.
ii. Symmetric.
iii. Transitive.

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Q 16Solve the Following Question.(5 Marks)5 Marks

Show that the relation $R,$ defined on the set A of all polygons as $R = \{(P_1, P_2): P_1$ and $P_2$ have same number of sides$\}$, is an equivalence relation. What is the set of all elements in A related to the right angle triangle $T$ with sides $3, 4$ and $5?$

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Q 17Solve the Following Question.(5 Marks)5 Marks

Let $Z$ be the set of all integers and $Z_0$ be the set of all non-zero integers. Let a relation $R$ on $Z \times Z_0$ be defined as $(a, b)R(c, d) ⇔ ad = bc$ for all $(a, b), (c, d) \in Z \times Z_0,$ Prove that $R$ is an equivalence relation on $Z \times Z_0.$

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Q 18Solve the Following Question.(5 Marks)5 Marks

Let R be the relation defined on the set A = {1, 2, 3, 4, 5, 6, 7} by R = {(a, b): both a and b are either odd or even}. Show that R is an equivalence relation. Further, show that all the elements of the subset {1, 3, 5, 7} are related to each other and all the elements of the subset {2, 4, 6} are related to each other, but no element of the subset {1, 3, 5, 7} is related to any element of the subset {2, 4, 6}.

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Q 19Solve the Following Question.(5 Marks)5 Marks

m is said to be related to n if m and n are integers and m - n is divisible by 13. Does this define an equivalence relation?

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Q 20Solve the Following Question.(5 Marks)5 Marks

Let n be a fixed positive integer. Define a relation R on Z as follows:
$(\text{a, b})\in\text{R}\Leftrightarrow\ \text{a}-\text{b}$ is divisible by n. Show that R is an equivalence relation on Z.

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Relations Maths - Relations

Relations question types

Relations questions

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