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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.
Solution :
Step 1: Defining the Relation and Set
Let $A$ be the set of all points in a Cartesian plane. The relation $R$ is defined on $A$ as: $R = \{(P, Q) \in A \times A : dist(O, P) = dist(O, Q)\}$, where $O$ is the origin $(0, 0)$. Let $d(X)$ denote the Euclidean distance of a point $X$ from the origin, $d(X) = \sqrt{x^2 + y^2}$. Thus, $R = \{(P, Q) : d(P) = d(Q)\}$.
Step 2: Proving Reflexivity
For any point $P \in A$, the distance of $P$ from the origin is $d(P)$. Clearly, $d(P) = d(P)$. [By the Reflexive Property of Equality]. Therefore, $(P, P) \in R$ for all $P \in A$. Hence, $R$ is reflexive.
Step 3: Proving Symmetry
Suppose $(P, Q) \in R$. By definition, $d(P) = d(Q)$. By the Symmetric Property of Equality, $d(Q) = d(P)$. This implies that $(Q, P) \in R$. Since $(P, Q) \in R \implies (Q, P) \in R$, $R$ is symmetric.
Step 4: Proving Transitivity
Suppose $(P, Q) \in R$ and $(Q, S) \in R$. Then, $d(P) = d(Q)$ and $d(Q) = d(S)$. By the Transitive Property of Equality, $d(P) = d(S)$. Therefore, $(P, S) \in R$. Since $R$ is reflexive, symmetric, and transitive, it is an equivalence relation [Per the Definition of Equivalence Relation].
Step 5: Geometric Interpretation of the Equivalence Class
Let $P = (x_0, y_0)$ be a point such that $P \neq (0, 0)$. The set of all points $Q(x, y)$ related to $P$ is the set $\{Q : d(Q) = d(P)\}$. Let $k = d(P) = \sqrt{x_0^2 + y_0^2}$. The condition $d(Q) = k$ becomes $\sqrt{x^2 + y^2} = k$, which squares to $x^2 + y^2 = k^2$. [By the standard equation of a circle with center $(0,0)$ and radius $r = k$]. This is a circle centered at the origin passing through $P$.
Step 6: Conclusion
The set of all points $Q$ related to $P$ is the set of points equidistant from the origin. Geometrically, this locus describes a circle with radius $r = \sqrt{x_0^2 + y_0^2}$ centered at the origin, which necessarily passes through $P$.
Final Solution: The relation R is an equivalence relation because it satisfies reflexivity, symmetry, and transitivity. The set of all points related to $P \neq (0,0)$ is a circle centered at the origin with radius equal to the distance of $P$ from the origin.
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(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 $x\}$
- 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.
- 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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