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Q1(iii):
Determine whether each of the following relations are reflexive, symmetric and transitive: (iii) Relation $R$ in the set $A = \{1, 2, 3, 4, 5, 6\}$ as $R = \{(x, y) : y$ is divisible by $x\}$
Solution :
Step 1: Formal Definition of the Relation
The set $A$ is defined as $A = \{1, 2, 3, 4, 5, 6\}$. The relation $R$ on set $A$ is defined as $R = \{(x, y) : y \text{ is divisible by } x\}$, where $x, y \in A$.
By enumerating the ordered pairs $(x, y)$ such that $x$ divides $y$ without remainder, we obtain the set $R$:
$R = \{ (1,1), (1,2), (1,3), (1,4), (1,5), (1,6), (2,2), (2,4), (2,6), (3,3), (3,6), (4,4), (5,5), (6,6) \}$
Step 2: Testing for Reflexivity
A relation $R$ is reflexive if for every $a \in A$, $(a, a) \in R$. [Definition of Reflexive Relation].
- For all $a \in \{1, 2, 3, 4, 5, 6\}$, $a$ is divisible by $a$ because $\frac{a}{a} = 1$ (an integer).
- Specifically, $(1,1), (2,2), (3,3), (4,4), (5,5), (6,6) \in R$.
Conclusion: Since $\forall a \in A, (a, a) \in R$, the relation $R$ is reflexive.
Step 3: Testing for Symmetry
A relation $R$ is symmetric if $(x, y) \in R \implies (y, x) \in R$ for all $x, y \in A$. [Definition of Symmetric Relation].
- Consider the element $(1, 2) \in R$ because $2$ is divisible by $1$.
- Test the converse: Is $(2, 1) \in R$? For $(2, 1)$ to be in $R$, $1$ must be divisible by $2$. Since $\frac{1}{2} = 0.5 \notin \mathbb{Z}$, $1$ is not divisible by $2$.
- Thus, $(1, 2) \in R$ but $(2, 1) \notin R$.
Conclusion: Since there exists at least one pair $(x, y) \in R$ such that $(y, x) \notin R$, the relation $R$ is not symmetric.
Step 4: Testing for Transitivity
A relation $R$ is transitive if $(x, y) \in R$ and $(y, z) \in R \implies (x, z) \in R$ for all $x, y, z \in A$. [Definition of Transitive Relation].
- Let $(x, y) \in R \implies y = k_1 x$ for some integer $k_1$.
- Let $(y, z) \in R \implies z = k_2 y$ for some integer $k_2$.
- Substituting the first into the second: $z = k_2 (k_1 x) = (k_1 k_2) x$.
- Since $k_1, k_2$ are integers, their product $(k_1 k_2)$ is also an integer.
- Therefore, $z$ is divisible by $x$, implying $(x, z) \in R$.
Example verification: $(2, 4) \in R$ and $(4, 4) \in R \implies (2, 4) \in R$. Or $(1, 2) \in R$ and $(2, 6) \in R \implies (1, 6) \in R$.
Conclusion: Since the condition holds for all triplets, the relation $R$ is transitive.
Step 5: Visual Representation
Final Solution: The relation $R$ is reflexive and transitive, but not symmetric.
More Questions from Class 12 Mathematics Relations and Functions EXERCISE 1.1
- Q1(i): Determine whether each of the following relations are reflexive, symmetric and transitive: (i) Relation $R$ in the set $A = \{1, 2, 3, ..., 13, 14\}$ defined as $R = \{(x, y) : 3x – y = 0\}$
- Q1(ii): Determine whether each of the following relations are reflexive, symmetric and transitive: (ii) Relation $R$ in the set $N$ of natural numbers defined as $R = \{(x, y) : y = x + 5$ and $x < 4\}$
- Q1(iv): Determine whether each of the following relations are reflexive, symmetric and transitive: (iv) Relation $R$ in the set $Z$ of all integers defined as $R = \{(x, y) : x – y$ is an integer\}$
- Q1(v)(a): Determine whether each of the following relations are reflexive, symmetric and transitive: (v) Relation $R$ in the set $A$ of human beings in a town at a particular time given by (a) $R = \{(x, y) : x$ and $y$ work at the same place\}
- Q1(v)(b): Determine whether each of the following relations are reflexive, symmetric and transitive: (v) Relation $R$ in the set $A$ of human beings in a town at a particular time given by (b) $R = \{(x, y) : x$ and $y$ live in the same locality\}
- Q1(v)(c): Determine whether each of the following relations are reflexive, symmetric and transitive: (v) Relation $R$ in the set $A$ of human beings in a town at a particular time given by (c) $R = \{(x, y) : x$ is exactly 7 cm taller than $y\}$
- Q1(v)(d): Determine whether each of the following relations are reflexive, symmetric and transitive: (v) Relation $R$ in the set $A$ of human beings in a town at a particular time given by (d) $R = \{(x, y) : x$ is wife of $y\}$
- Q1(v)(e): Determine whether each of the following relations are reflexive, symmetric and transitive: (v) Relation $R$ in the set $A$ of human beings in a town at a particular time given by (e) $R = \{(x, y) : x$ is father of $y\}$
- Q2: Show that the relation $R$ in the set $R$ of real numbers, defined as $R = \{(a, b) : a \leq b^2\}$ is neither reflexive nor symmetric nor transitive.
- Q3: Check whether the relation $R$ defined in the set $\{1, 2, 3, 4, 5, 6\}$ as $R = \{(a, b) : b = a + 1\}$ is reflexive, symmetric or transitive.
- Q4: Show that the relation $R$ in $R$ defined as $R = \{(a, b) : a \leq b\}$, is reflexive and transitive but not symmetric.
- Q5: Check whether the relation $R$ in $R$ defined by $R = \{(a, b) : a \leq b^3\}$ is reflexive, symmetric or transitive.
- Q6: Show that the relation $R$ in the set $\{1, 2, 3\}$ given by $R = \{(1, 2), (2, 1)\}$ is symmetric but neither reflexive nor transitive.
- Q7: Show that the relation $R$ in the set $A$ of all the books in a library of a college, given by $R = \{(x, y) : x$ and $y$ have same number of pages\}$ is an equivalence relation.
- Q8: Show that the relation $R$ in the set $A = \{1, 2, 3, 4, 5\}$ given by $R = \{(a, b) : |a – b|$ is even\}, is an equivalence relation. Show that all the elements of $\{1, 3, 5\}$ are related to each other and all the elements of $\{2, 4\}$ are related to each other. But no element of $\{1, 3, 5\}$ is related to any element of $\{2, 4\}$.
- Q9(i): Show that each of the relation $R$ in the set $A = \{x \in Z : 0 \leq x \leq 12\}$, given by (i) $R = \{(a, b) : |a – b|$ is a multiple of 4\} is an equivalence relation. Find the set of all elements related to 1 in each case.
- Q9(ii): Show that each of the relation $R$ in the set $A = \{x \in Z : 0 \leq x \leq 12\}$, given by (ii) $R = \{(a, b) : a = b\} is an equivalence relation. Find the set of all elements related to 1 in each case.
- Q10(i): Give an example of a relation. Which is (i) Symmetric but neither reflexive nor transitive.
- Q10(ii): Give an example of a relation. Which is (ii) Transitive but neither reflexive nor symmetric.
- Q10(iii): Give an example of a relation. Which is (iii) Reflexive and symmetric but not transitive.
- Q10(iv): Give an example of a relation. Which is (iv) Reflexive and transitive but not symmetric.
- Q10(v): Give an example of a relation. Which is (v) Symmetric and transitive but not reflexive.
- Q11: Show that the relation $R$ in the set $A$ of points in a plane given by $R = \{(P, Q) :$ distance of the point $P$ from the origin is same as the distance of the point $Q$ from the origin\}, is an equivalence relation. Further, show that the set of all points related to a point $P \neq (0, 0)$ is the circle passing through $P$ with origin as centre.
- Q12: Show that the relation $R$ defined in the set $A$ of all triangles as $R = \{(T_1, T_2) : T_1$ is similar to $T_2\}$, is equivalence relation. Consider three right angle triangles $T_1$ with sides 3, 4, 5, $T_2$ with sides 5, 12, 13 and $T_3$ with sides 6, 8, 10. Which triangles among $T_1$, $T_2$ and $T_3$ are related?
- Q13: Show that the relation $R$ defined in 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?
- Q14: Let $L$ be the set of all lines in XY plane and $R$ be the relation in $L$ defined as $R = \{(L_1, L_2) : L_1$ is parallel to $L_2\}$. Show that $R$ is an equivalence relation. Find the set of all lines related to the line $y = 2x + 4$.
- Q15: Let $R$ be the relation in the set $\{1, 2, 3, 4\}$ given by $R = \{(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)\}$. Choose the correct answer.
- Q16: Let $R$ be the relation in the set $N$ given by $R = \{(a, b) : a = b – 2, b > 6\}$. Choose the correct answer.
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