Determine whether each of the following relations are reflexive, symmetric and transitive:
(i) Neither reflexive nor symmetric nor transitive.
(ii) Neither reflexive nor symmetric but transitive.
(iii) Reflexive and transitive but not symmetric.
(iv) Reflexive, symmetric and transitive.
(v)(a) Reflexive, symmetric and transitive.
(v)(b) Reflexive, symmetric and transitive.
(v)(c) Neither reflexive nor symmetric nor transitive.
(v)(d) Neither reflexive nor symmetric but transitive.
(v)(e) Neither reflexive nor symmetric nor transitive.
Show that the relation \(R\) in the set \(\mathbb{R}\) of real numbers, defined as \(R = \{(a,b) : a \leq b^2\}\), is neither reflexive nor symmetric nor transitive.
The relation \(R = \{(a,b) : a \leq b^2\}\) is neither reflexive nor symmetric nor transitive.
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.
Neither reflexive nor symmetric nor transitive.
Show that the relation \(R\) in \(\mathbb{R}\) defined as \(R = \{(a,b) : a \leq b\}\) is reflexive and transitive but not symmetric.
The relation \(R = \{(a,b) : a \leq b\}\) is reflexive and transitive but not symmetric.
Check whether the relation \(R\) in \(\mathbb{R}\) defined by \(R = \{(a,b) : a \leq b^3\}\) is reflexive, symmetric or transitive.
Neither reflexive nor symmetric nor transitive.
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.
The relation \(R = \{(1,2), (2,1)\}\) is symmetric but neither reflexive nor transitive.
Show that the relation \(R\) in the set \(A\) of all the books in a library of a college, given by \(R = \{(x,y) : x \text{ and } y \text{ have same number of pages}\}\), is an equivalence relation.
The relation \(R = \{(x,y) : x \text{ and } y \text{ have same number of pages}\}\) is an equivalence relation (reflexive, symmetric and transitive).
Show that the relation \(R\) in the set \(A = \{1,2,3,4,5\}\) given by \(R = \{(a,b) : |a - b| \text{ is even}\}\) 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 \(\{2,4\}\).
The relation \(R = \{(a,b) : |a - b| \text{ is even}\}\) on \(A = \{1,2,3,4,5\}\) is an equivalence relation. The elements \(1,3,5\) form one equivalence class and \(2,4\) form another; no element of \(\{1,3,5\}\) is related to any element of \(\{2,4\}\).
Show that each of the relations \(R\) in the set \(A = \{x \in \mathbb{Z} : 0 \leq x \leq 12\}\), given by
is an equivalence relation. Find the set of all elements related to \(1\) in each case.
Both (i) and (ii) define equivalence relations on \(A\). The set of all elements related to \(1\) is:
(i) \(\{1,5,9\}\)
(ii) \(\{1\}\)
Give an example of a relation which is:
Examples may vary. One possible set of examples is:
(i) On \(A = \{1,2,3\}\), let \(R = \{(1,2), (2,1)\}\).
(ii) On \(A = \mathbb{Z}\), let \(R = \{(a,b) : a < b\}\).
(iii) On \(A = \mathbb{R}\), let \(R = \{(a,b) : a = b \text{ or } a = -b\}\).
(iv) On \(A = \mathbb{R}\), let \(R = \{(a,b) : a \leq b\}\).
(v) On \(A = \mathbb{R}\), let \(R = \{(a,b) : a = b = 0\}\).
Show that the relation \(R\) in the set \(A\) of points in a plane given by \(R = \{(P,Q) : \text{distance of the point } P \text{ from the origin is same as the distance of the point } Q \text{ from the origin}\}\) 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.
The relation \(R\) defined by equality of distance from the origin is an equivalence relation (reflexive, symmetric and transitive). For a fixed point \(P \neq (0,0)\), the set of all points related to \(P\) is the circle with centre at the origin and radius equal to the distance of \(P\) from the origin; this circle passes through \(P\).
Show that the relation \(R\) defined in the set \(A\) of all triangles as \(R = \{(T_1,T_2) : T_1 \text{ is similar to } T_2\}\) is an 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?
The relation \(R\) "is similar to" on the set of all triangles is an equivalence relation. Among the given triangles, \(T_1\) (sides \(3,4,5\)) is similar to \(T_3\) (sides \(6,8,10\)), so \(T_1\) is related to \(T_3\).
Show that the relation \(R\) defined in the set \(A\) of all polygons as \(R = \{(P_1,P_2) : P_1 \text{ and } P_2 \text{ have same number of sides}\}\) is an equivalence relation. What is the set of all elements in \(A\) related to the right angled triangle \(T\) with sides \(3,4\) and \(5\)?
The relation \(R\) is an equivalence relation. The set of all elements in \(A\) related to the right angled triangle \(T\) is the set of all triangles (all polygons having three sides).
Let \(L\) be the set of all lines in the \(XY\)-plane and \(R\) be the relation in \(L\) defined as \(R = \{(L_1,L_2) : L_1 \text{ is parallel to } L_2\}\). Show that \(R\) is an equivalence relation. Find the set of all lines related to the line \(y = 2x + 4\).
The relation \(R\) "is parallel to" is an equivalence relation. The set of all lines related to the line \(y = 2x + 4\) is the set of all lines of the form \(y = 2x + c\), where \(c \in \mathbb{R}\).
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.
(A) \(R\) is reflexive and symmetric but not transitive.
(B) \(R\) is reflexive and transitive but not symmetric.
(C) \(R\) is symmetric and transitive but not reflexive.
(D) \(R\) is an equivalence relation.
(B) \(R\) is reflexive and transitive but not symmetric.
Let \(R\) be the relation in the set \(\mathbb{N}\) given by \(R = \{(a,b) : a = b - 2, b > 6\}\). Choose the correct answer.
(A) \((2,4) \in R\)
(B) \((3,8) \in R\)
(C) \((6,8) \in R\)
(D) \((8,7) \in R\)
(C) \((6,8) \in R\).