Advertisements
Advertisements
Question
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
Solution
(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
RELATED QUESTIONS
Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation.
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.
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 father of and y}
Test whether the following relation R3 is (i) reflexive (ii) symmetric and (iii) transitive:
R3 on R is defined by (a, b) ∈ R3 `⇔` a2 – 4ab + 3b2 = 0.
The following relation is defined on the set of real numbers.
aRb if a – b > 0
Find whether relation is reflexive, symmetric or transitive.
If A = {1, 2, 3, 4} define relations on A which have properties of being symmetric but neither reflexive nor transitive ?
Let R be a relation defined on the set of natural numbers N as
R = {(x, y) : x, y ∈ N, 2x + y = 41}
Find the domain and range of R. Also, verify whether R is (i) reflexive, (ii) symmetric (iii) transitive.
Show that the relation '≥' on the set R of all real numbers is reflexive and transitive but not symmetric ?
Give an example of a relation which is reflexive and transitive but not symmetric?
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?
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.
Show that the relation R, defined in the set A of all polygons as R = {(P1, P2) : P1 and P2 have the 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?
Write the smallest reflexive relation on set A = {1, 2, 3, 4}.
Let R be the equivalence relation on the set Z of the integers given by R = { (a, b) : 2 divides a - b }.
Write the equivalence class [0].
Let the relation R be defined on N by aRb iff 2a + 3b = 30. Then write R as a set of ordered pairs
Let R be the relation over the set of all straight lines in a plane such that l1 R l2 ⇔ l 1⊥ l2. Then, R is _____________ .
If A = {a, b, c}, then the relation R = {(b, c)} on A 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 ______________ .
A relation R is defined from {2, 3, 4, 5} to {3, 6, 7, 10} by : x R y ⇔ x is relatively prime to y. Then, domain of R is ______________ .
A relation ϕ from C to R is defined by x ϕ y ⇔ | x | = y. Which one is correct?
If A = {a, b, c, d}, then a relation R = {(a, b), (b, a), (a, a)} on A is _____________ .
Let R be the relation on the set A = {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Then, _____________________ .
Mark the correct alternative in the following question:
For real numbers x and y, define xRy if `x-y+sqrt2` is an irrational number. Then the relation R is ___________ .
Show that the relation R on the set Z of integers, given by R = {(a,b):2divides (a - b)} is an equivalence relation.
If A = {a, b, c}, B = (x , y} find B × A.
If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is ______.
Let the relation R be defined in N by aRb if 2a + 3b = 30. Then R = ______.
Let the relation R be defined on the set A = {1, 2, 3, 4, 5} by R = {(a, b) : |a2 – b2| < 8. Then R is given by ______.
Every relation which is symmetric and transitive is also reflexive.
R = {(1, 1), (2, 2), (1, 2), (2, 1), (2, 3)} be a relation on A, then R is ____________.
If f(x) = `1 - 1/"x", "then f"("f"(1/"x"))` ____________.
Let R be a relation on the set N of natural numbers denoted by nRm ⇔ n is a factor of m (i.e. n | m). Then, R is ____________.
A relation R in set A = {1, 2, 3} is defined as R = {(1, 1), (1, 2), (2, 2), (3, 3)}. Which of the following ordered pair in R shall be removed to make it an equivalence relation in A?
Sherlin and Danju are playing Ludo at home during Covid-19. While rolling the dice, Sherlin’s sister Raji observed and noted the possible outcomes of the throw every time belongs to set {1,2,3,4,5,6}. Let A be the set of players while B be the set of all possible outcomes.
A = {S, D}, B = {1,2,3,4,5,6}
- Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible?
The relation > (greater than) on the set of real numbers is
On the set N of all natural numbers, define the relation R by a R b, if GCD of a and b is 2. Then, R is
A relation 'R' in a set 'A' is called a universal relation, if each element of' A' is related to :-
Let R1 and R2 be two relations defined as follows :
R1 = {(a, b) ∈ R2 : a2 + b2 ∈ Q} and
R2 = {(a, b) ∈ R2 : a2 + b2 ∉ Q}, where Q is the set of all rational numbers. Then ______
