Advertisements
Advertisements
प्रश्न
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
उत्तर
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
संबंधित प्रश्न
Determine whether the following relation is reflexive, symmetric and transitive:
Relation R in the set N of natural numbers defined as R = {(x, y) : y = x + 5 and x < 4}.
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}
Given an example of a relation. Which is transitive but neither reflexive nor symmetric.
Show that the relation R defined in the set A of all polygons as R = {(P1, P2): P1 and P2 have 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 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}
Three relations R1, R2 and R3 are defined on a set A = {a, b, c} as follows:
R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}
R2 = {(a, a)}
R3 = {(b, c)}
R4 = {(a, b), (b, c), (c, a)}.
Find whether or not each of the relations R1, R2, R3, R4 on A is (i) reflexive (ii) symmetric and (iii) transitive.
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.
Give an example of a relation which is reflexive and transitive but not symmetric?
Defines a relation on N:
x + 4y = 10, x, y ∈ N
Determine the above relation is reflexive, symmetric and transitive.
Prove that the relation R on Z defined by
(a, b) ∈ R ⇔ a − b is divisible by 5
is an equivalence relation on Z.
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.
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 ?
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 A = {0, 1, 2, 3} and R be a relation on A defined as
R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}
Is R reflexive? symmetric? transitive?
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 a relation on the set N given by
R = {(a, b) : a = b − 2, b > 6}. Then,
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 A = {a, b, c}, B = (x , y} find B × A.
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 map which is not one-one but onto
The maximum number of equivalence relations on the set A = {1, 2, 3} are ______.
If f(x) = `1 - 1/"x", "then f"("f"(1/"x"))` ____________.
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.
- Ravi wishes to form all the relations possible from B to G. How many such relations are possible?
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 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 ______
Which relation is called an Empty Relation?
What characterizes an Identity Relation?
Which relation is reflexive on \(A=\{1,2\}\)?
Which pair of relations are called trivial relations?
