Advertisements
Advertisements
Question
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
Solution
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
RELATED QUESTIONS
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 = {1, 2, 3}. Then number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is ______.
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 live in the same locality}
Let A = {1, 2, 3}, and let R1 = {(1, 1), (1, 3), (3, 1), (2, 2), (2, 1), (3, 3)}, R2 = {(2, 2), (3, 1), (1, 3)}, R3 = {(1, 3), (3, 3)}. Find whether or not each of the relations R1, R2, R3 on A is (i) reflexive (ii) symmetric (iii) transitive.
The following relation is defined on the set of real numbers. aRb if |a| ≤ b
Find whether relation is reflexive, symmetric or transitive.
If A = {1, 2, 3, 4} define relations on A which have properties of being reflexive, transitive but not symmetric ?
If A = {1, 2, 3, 4} define relations on A which have properties of being symmetric but neither reflexive nor transitive ?
Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons.
Show that the relation '≥' on the set R of all real numbers is reflexive and transitive but not symmetric ?
Defines a relation on N:
xy is square of an integer, x, y ∈ N
Determine the above relation is reflexive, symmetric and transitive.
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?
If R and S are transitive relations on a set A, then prove that R ∪ S may not be a transitive relation on A.
Write the identity relation on set A = {a, b, c}.
If R is a symmetric relation on a set A, then write a relation between R and R−1.
Let R = {(x, y) : |x2 − y2| <1) be a relation on set A = {1, 2, 3, 4, 5}. Write R as a set of ordered pairs.
Let the relation R be defined on the set A = {1, 2, 3, 4, 5} by R = {(a, b) : | a2- b2 | < 8}. Write R as a set of ordered pairs.
R is a relation on the set Z of integers and it is given by
(x, y) ∈ R ⇔ | x − y | ≤ 1. Then, R is ______________ .
The relation R defined on the set A = {1, 2, 3, 4, 5} by
R = {(a, b) : | a2 − b2 | < 16} is given by ______________ .
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 _____________ .
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 ______.
Let R be a relation on N defined by x + 2y = 8. The domain of R is _______________ .
In the set of natural numbers N, define a relation R as follows: ∀ n, m ∈ N, nRm if on division by 5 each of the integers n and m leaves the remainder less than 5, i.e. one of the numbers 0, 1, 2, 3 and 4. Show that R is equivalence relation. Also, obtain the pairwise disjoint subsets determined by R
Let n be a fixed positive integer. Define a relation R in Z as follows: ∀ a, b ∈ Z, aRb if and only if a – b is divisible by n. Show that R is an equivalance relation
If A = {1, 2, 3, 4 }, define relations on A which have properties of being:
reflexive, transitive but not symmetric
The following defines a relation on N:
x + 4y = 10 x, y ∈ N.
Determine which of the above relations are reflexive, symmetric and transitive.
Let A = { 2, 3, 6 } Which of the following relations on A are reflexive?
Let A = {1, 2, 3} and R = {(1, 2), (2, 3), (1, 3)} be a relation on A. Then, R is ____________.
If A is a finite set containing n distinct elements, then the number of relations on A is equal to ____________.
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 ____________.
Total number of equivalence relations defined in the set S = {a, b, c} is ____________.
The relation R is defined on the set of natural numbers as {(a, b) : a = 2b}. Then, R-1 is given by ____________.
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 relation R be defined by R = {(L1, L2): L1║L2 where L1, L2 ∈ L} then R is ____________ relation.
If A = {1,2,3}, B = {4,6,9} and R is a relation from A to B defined by ‘x is smaller than y’. The range of R is ____________.
The relation R = {(1,1),(2,2),(3,3)} on {1,2,3} 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?
Let R = {(a, b): a = a2} for all, a, b ∈ N, then R salifies.
Let a set A = A1 ∪ A2 ∪ ... ∪ Ak, where Ai ∩ Aj = Φ for i ≠ j, 1 ≤ i, j ≤ k. Define the relation R from A to A by R = {(x, y): y ∈ Ai if and only if x ∈ Ai, 1 ≤ i ≤ k}. Then, R is ______.
If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.
