Advertisements
Advertisements
Question
Show that the relation R defined by (a, b)R(c,d) ⇒ a + d = b + c on the A x A , where A = {1, 2,3,...,10} is an equivalence relation. Hence write the equivalence class [(3, 4)]; a, b, c,d ∈ A.
Advertisements
Solution
Here (a, b)R(c,d) ⇒ a + d = b + c on A x A, where A = {1, 2,3,...,10} .
Reflexivity: Let (a, b) be an arbitrary element of A x A. Then, (a,b) ∈ A x A `forall` a, b ∈ A.
So, a + b = b + a
⇒ (a,b) R (a,b).
Thus, (a,b) R (a,b) `forall` (a,b) ∈ A x A.
Hence R is reflexive.
Symmetry: Let (a,b), (c,d) ∈ A x A be such that (a,b) R (c,d).
Then, a + d = b + c
⇒ c + b = d + a
⇒ (c,d ) R (a,b).
Thus, (a,b) R (c,d)
⇒ (c,d) R (a,b) `forall` (a,b), (c,d) ∈ A x A.
Hence R is symmetric.
Transitivity: Let (a,b),(c,d),(e,f) ∈ A x A be such that (a,b) R (c,d) R (e,f).
Then, a + d = b + c and c + f = d + e
⇒ (a+d) + (c+f)
= (b + c) + (d+e)
⇒ a + f = b + e
⇒ (a, b) R (e,f).
That is (a,b) R (c,d) and (c,d) R (e,f)
⇒ (a,b) R (e,f) `forall` (a,b), (c,d), (e,f) ∈ A x A.
Hence R is transitive.
Since R is reflexive, symmetric and transitive so, R is an equivalence relation as well.
For the equivalence class of [(3, 4)], we need to find (a,b) s.t. (a,b) R (3,4)
⇒ a + 4 = b + 3
⇒ b - a = 1.
So, [(3,4)] = {(1,2),(2,3),(3,4),(4,5),(5,6),(6,7),(7,8),(8,9),(9,10)}.
APPEARS IN
RELATED QUESTIONS
Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b2} is 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.
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.
Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is the same as the distance of the point Q from the origin} is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with the origin as its centre.
Show that the relation R defined in the set A of all polygons as R = {(P1, P2): P1 and P2have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?
Let L be the set of all lines in the 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.
If A = {1, 2, 3, 4} define relations on A which have properties of being reflexive, transitive but not symmetric ?
An integer m is said to be related to another integer n if m is a multiple of n. Check if the relation is symmetric, reflexive and transitive.
Let A = {1, 2, 3} and R = {(1, 2), (1, 1), (2, 3)} be a relation on A. What minimum number of ordered pairs may be added to R so that it may become a transitive relation on A.
Show that the relation R on the set Z of integers, given by
R = {(a, b) : 2 divides a – b}, is an equivalence relation.
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.
Show that the relation R on 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.
Let A = {3, 5, 7}, B = {2, 6, 10} and R be a relation from A to B defined by R = {(x, y) : x and y are relatively prime}. Then, write R and R−1.
Let R = {(a, a3) : a is a prime number less than 5} be a relation. Find the range of R.
Let R = {(a, a), (b, b), (c, c), (a, b)} be a relation on set A = a, b, c. Then, R is _______________ .
If A = {1, 2, 3}, then a relation R = {(2, 3)} on A is _____________ .
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}. Find A × (B ∪ C).
Let Z be the set of integers and R be the relation defined in Z such that aRb if a – b is divisible by 3. Then R partitions the set Z into ______ pairwise disjoint subsets
Let R be relation defined on the set of natural number N as follows:
R = {(x, y): x ∈N, y ∈N, 2x + y = 41}. Find the domain and range of the relation R. Also verify whether R is reflexive, symmetric and transitive
The following defines a relation on N:
x y is square of an integer x, y ∈ N
Determine which of the above relations are reflexive, symmetric and transitive.
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 ______.
The maximum number of equivalence relations on the set A = {1, 2, 3} are ______.
Let A = { 2, 3, 6 } Which of the following relations on A are reflexive?
Let `"f"("x") = ("x" - 1)/("x" + 1),` then f(f(x)) is ____________.
The relation R is defined on the set of natural numbers as {(a, b) : a = 2b}. Then, R-1 is given by ____________.
A relation 'R' in a set 'A' is called a universal relation, if each element of' A' is related to :-
A relation 'R' in a set 'A' is called reflexive, if
Define the relation R in the set N × N as follows:
For (a, b), (c, d) ∈ N × N, (a, b) R (c, d) if ad = bc. Prove that R is an equivalence relation in N × N.
Let R = {(x, y) : x, y ∈ N and x2 – 4xy + 3y2 = 0}, where N is the set of all natural numbers. Then the relation R is ______.
If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.
