Advertisements
Advertisements
प्रश्न
Let Z be the set of all integers and Z0 be the set of all non-zero integers. Let a relation R on Z × Z0be defined as (a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z0,
Prove that R is an equivalence relation on Z × Z0.
Advertisements
उत्तर
We observe the following properties of R.
Reflexivity :
Let (a, b) be an arbitrary element of Z × Z0. Then,
(a, b) ∈ Z × Z0
⇒ a, b ∈ Z, Z0
⇒ ab = ba
⇒ (a, b) ∈ R for all (a, b) ∈ Z × Z0
So, R is reflexive on Z × Z0.
Symmetry :
Let (a, b), (c, d) ∈ Z×Z0 such that (a, b) R (c, d). Then,(a, b) R (c, d)
⇒ ad = bc
⇒ cb = da
⇒ (c, d) R (a, b)
Thus,
(a, b) R (c, d)⇒ (c, d) R (a, b) for all (a, b), (c, d) ∈ Z× Z0
So, R is symmetric on Z×Z0.
Transitivity :
Let(a,b), (c,d),(e,f) ∈N × N0suchthat a,b) R(c,d) and(c,d) R(e,f).Then,
a, b) R (c, d)⇒ ad =bc (c, d) R (e, f) ⇒ cf =de } ⇒ (ad) (cf)=(bc) (de)
⇒ af=be
⇒ (a, b) R (e, f)
Thus,
(a, b) R (c, d) and (c, d ) R (e, f) ⇒ (a, b) R (e, f)
⇒(a, b) R (e, f) for all values (a, b), (c, d), (e, f) ∈ N × N0
So, R is transitive on N × N0.
APPEARS IN
संबंधित प्रश्न
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 exactly 7 cm taller than y}
Show that the relation R in the set A = {x ∈ Z : 0 ≤ x ≤ 12} given by R = {(a, b) : a = b} is an equivalence relation. Find the set of all elements related to 1.
Given an example of a relation. Which is reflexive and transitive but not symmetric.
Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that R = {(a, b) : a, b ∈ A, |a – b| is divisible by 4}is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2]
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}
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.
Show that the relation R defined by R = {(a, b) : a – b is divisible by 3; a, b ∈ Z} is an equivalence relation.
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 L be the set of all lines in XY-plane and R be the relation in L defined as R = {L1, L2) : L1 is parallel to L2}. Show that R is an equivalence relation. Find the set of all lines related to the line y= 2x + 4.
Define an equivalence relation ?
Let R = {(a, a3) : a is a prime number less than 5} be a relation. Find the range of R.
Let A = {1, 2, 3}. Then, the number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is ______.
R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x − 3. Then, R−1 is ______________ .
Let A = {1, 2, 3} and B = {(1, 2), (2, 3), (1, 3)} be a relation on A. Then, R is ________________ .
S is a relation over the set R of all real numbers and it is given by (a, b) ∈ S ⇔ ab ≥ 0. Then, S is _______________ .
Mark the correct alternative in the following question:
Let A = {1, 2, 3} and consider the relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)}. Then, R is _______________ .
Mark the correct alternative in the following question:
The maximum number of equivalence relations on the set A = {1, 2, 3} is _______________ .
Show that the relation R on the set Z of integers, given by R = {(a,b):2divides (a - b)} is an equivalence relation.
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)]
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 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.
Let A = {1, 2, 3}. Which of the following is not an equivalence relation on A?
If A is a finite set containing n distinct elements, then the number of relations on A is equal to ____________.
Let `"f"("x") = ("x" - 1)/("x" + 1),` then f(f(x)) is ____________.
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 ____________.
Let R be the relation “is congruent to” on the set of all triangles in a plane is ____________.
An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.
Ravi decides to explore these sets for various types of relations and functions.
- Let R: B → B be defined by R = {(x, y): x and y are students of same sex}, Then this relation R is ____________.
The relation R = {(1,1),(2,2),(3,3)} on {1,2,3} is ____________.
There are 600 student in a school. If 400 of them can speak Telugu, 300 can speak Hindi, then the number of students who can speak both Telugu and Hindi is:
Which of the following is/are example of symmetric
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 ______
Let L be a set of all straight lines in a plane. The relation R on L defined as 'perpendicular to' is ______.
Statement 1: The intersection of two equivalence relations is always an equivalence relation.
Statement 2: The Union of two equivalence relations is always an equivalence relation.
Which one of the following is correct?
If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.
