Advertisements
Advertisements
प्रश्न
Let C be the set of all complex numbers and C0 be the set of all no-zero complex numbers. Let a relation R on C0 be defined as
`z_1 R z_2 ⇔ (z_1 -z_2)/(z_1 + z_2)` is real for all z1, z2 ∈ C0.
Show that R is an equivalence relation.
Advertisements
उत्तर
(i) Test for reflexivity:
Since, `(z_1 -z_1)/(z_1 + z_1)`= 0, which is a real number.
So, (z1, z1) ∈ R
Hence, R is relexive relation.
(ii) Test for symmetric:
Let ( z1, z2 ) ∈ R.
Then `(z_1 -z_2)/(z_1 + z_2) =x`, where x is real
⇒ − `(z_1 -z_2)/(z_1 + z_2) = -x `
⇒ `(z_2 -z_1)/(z_2 + z_1)` = −x, is also a real number
So, (z2, z1) ∈ R
Hence, R is symmetric relation.
(iii) Test for transivity:
Let (z1, z2) ∈ R and (z2, z3) ∈ R.
Then,
`(z_1 -z_2)/(z_1 + z_2) x,`where x is a real number.
⇒ z1 − z2 = xz1 + xz2
⇒ z1 − xz1 = z2 + xz2
⇒ z1 (1 − x) = z2 (1 + x)
⇒ `z_1/z_2 = (1 +x )/(1-x)` ...(1)
Also,
`(z_2 -z_3)/(z_2+ z_3)`= y, where y is a real number.
⇒ z2 − z3= yz2 + Yz3
⇒z2 − yz2 = z3 + yz3
⇒ z2 (1 − y) = z3 (1 + y)
⇒ `z_2/z_3 = ((1+y))/((1 -y))` ...(2)
Dividing (1) and (2), we get
`z_1/z_3= ((1+x)/(1-x)) xx ((1-y)/(1 +y))` = z, where z is a real number.
`(z_1 -z_3)/(z_1+ z_3) = (z-1 ) /(z+1), which is real `
⇒ (z1, z3) ∈ R
Hence, R is transitive relation
From (i), (ii), and (iii),
R is an equivalenve relation.
APPEARS IN
संबंधित प्रश्न
Determine whether the following relation is reflexive, symmetric and transitive:
Relation R in the set A = {1, 2, 3, ..., 13, 14} defined as R = {(x, y) : 3x – y = 0}.
Determine whether the following relation is reflexive, symmetric and transitive:
Relation R in the set Z of all integers defined as R = {(x, y) : x – y is an integer}.
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}.
Given an example of a relation. Which is symmetric but neither reflexive nor transitive.
Given an example of a relation. Which is reflexive and symmetric but not transitive.
Let A be the set of all human beings in a town at a particular time. Determine whether the following relation is reflexive, symmetric and transitive:
R = {(x, y) : x and y work at the same place}
Let A be the set of all human beings in a town at a particular time. Determine whether the following relation is reflexive, symmetric and transitive:
R = {(x, y) : x is wife of y}
If A = {1, 2, 3, 4} define relations on A which have properties of being symmetric but neither reflexive nor transitive ?
Give an example of a relation which is reflexive and symmetric but not transitive?
Give an example of a relation which is symmetric and transitive but not reflexive?
Let Z be the set of integers. Show that the relation
R = {(a, b) : a, b ∈ Z and a + b is even}
is an equivalence relation on Z.
Let R be a relation on the set A of ordered pair of integers defined by (x, y) R (u, v) if xv = yu. Show that R is an equivalence relation.
If R and S are relations on a set A, then prove that R and S are symmetric ⇒ R ∩ S and R ∪ S are symmetric ?
Define an equivalence relation ?
R is a relation on the set Z of integers and it is given by
(x, y) ∈ R ⇔ | x − y | ≤ 1. Then, R is ______________ .
If A = {1, 2, 3}, B = {1, 4, 6, 9} and R is a relation from A to B defined by 'x is greater than y'. The range of R is ______________ .
If R is a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)}, then R is ____________ .
If `f(x) = (4x + 3)/(6x - 4), x ≠ 2/3`, show that fof (x) = x for all `x ≠ 2/3`. Also, find the inverse of f.
Let A = {0, 1, 2, 3} and define a relation R on A as follows: R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R reflexive? symmetric? transitive?
Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if and only if l is perpendicular to m ∀ l, m ∈ L. Then R is ______.
Let A = {1, 2, 3, ... 9} and R be the relation in A × A defined by (a, b) R(c, d) if a + d = b + c for (a, b), (c, d) in A × A. Prove that R is an equivalence relation and also obtain the equivalent class [(2, 5)]
Let T be the set of all triangles in the Euclidean plane, and let a relation R on T be defined as aRb if a is congruent to b ∀ a, b ∈ T. Then R is ______.
Consider the non-empty set consisting of children in a family and a relation R defined as aRb if a is brother of b. Then R is ______.
Let us define a relation R in R as aRb if a ≥ b. Then R is ______.
Let A = {1, 2, 3, 4, 5, 6} Which of the following partitions of A correspond to an equivalence relation on A?
Let A = {x : -1 ≤ x ≤ 1} and f : A → A is a function defined by f(x) = x |x| then f is ____________.
Let the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12}, given by R = {(a, b) : |a – b| is a multiple of 4}. Then [1], the equivalence class containing 1, is:
Given triangles with sides T1: 3, 4, 5; T2: 5, 12, 13; T3: 6, 8, 10; T4: 4, 7, 9 and a relation R inset of triangles defined as R = `{(Delta_1, Delta_2) : Delta_1 "is similar to" Delta_2}`. Which triangles belong to the same equivalence class?
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- Let R = `{ ("L"_1, "L"_2) ∶ "L"_1 bot "L"_2 "where" "L"_1, "L"_2 in "L" }` which of the following is true?
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- Let R = {(L1, L2 ): L1 is parallel to L2 and L1: y = x – 4} then which of the following can be taken as L2?
The relation R = {(1,1),(2,2),(3,3)} on {1,2,3} is ____________.
If A is a finite set consisting of n elements, then the number of reflexive relations on A is
The relation > (greater than) on the set of real numbers is
In a group of 52 persons, 16 drink tea but not coffee, while 33 drink tea. How many persons drink coffee but not tea?
A relation 'R' in a set 'A' is called a universal relation, if each element of' A' is related to :-
If f(x + 2a) = f(x – 2a), then f(x) is:
Let N be the set of all natural numbers and R be a relation on N × N defined by (a, b) R (c, d) `⇔` ad = bc for all (a, b), (c, d) ∈ N × N. Show that R is an equivalence relation on N × N. Also, find the equivalence class of (2, 6), i.e., [(2, 6)].
