Advertisements
Advertisements
प्रश्न
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.
Advertisements
उत्तर
We observe the following properties of R.
Reflexivity :
Let a be an arbitrary element of Z. Then,
a ∈ R
Clearly, a+a = 2a is even for all a ∈ Z.
⇒ (a, a) ∈ R for all a ∈ Z
So, R is reflexive on Z.
Symmetry :
Let (a, b) ∈ R
⇒ a+b is even
⇒ b+a is even
⇒ (b, a) ∈ R for all a, b ∈ Z
So, R is symmetric on Z.
Transitivity :
Let (a, b) and (b, c) ∈ R
⇒ a+b and b+c are even
Now, let a+b = 2x for some x ∈ Z
and b+c = 2y for some y ∈ Z
Adding the above two, we get
a+2b +c = 2x + 2y
⇒ a+c = 2 (x+y−b), which is even for all x, y, b ∈ Z
Thus, (a, c) ∈ R
So, R is transitive on Z.
Hence, R is an equivalence relation on Z
APPEARS IN
संबंधित प्रश्न
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. Also, obtain the equivalence class [(2, 5)].
Determine whether the following relation is reflexive, symmetric and transitive:
Relation R in the set A = {1, 2, 3, 4, 5, 6} as R = {(x, y) : y is divisible by x}.
Determine whether the following relation is reflexive, symmetric and transitive:
Relation R in the set Z of all integers defined as R = {(x, y) : x – y is an integer}.
Show that the relation R defined in the set A of all triangles as R = {(T1, T2) : T1 is similar to T2}, is an equivalence relation. Consider three right angle triangles T1 with sides 3, 4, 5, T2 with sides 5, 12, 13 and T3 with sides 6, 8, and 10. Which triangles among T1, T2 and T3 are related?
Test whether the following relation R2 is (i) reflexive (ii) symmetric and (iii) transitive:
R2 on Z defined by (a, b) ∈ R2 ⇔ |a – b| ≤ 5
The following relation is defined on the set of real numbers.
aRb if a – b > 0
Find whether relation is reflexive, symmetric or 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.
Is it true that every relation which is symmetric and transitive is also reflexive? Give reasons.
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.
Defines a relation on N :
x > y, x, y ∈ N
Determine the above relation is reflexive, symmetric and transitive.
Defines a relation on N :
x + y = 10, x, y∈ N
Determine the above relation is reflexive, symmetric and transitive.
Defines a relation on N:
x + 4y = 10, x, y ∈ N
Determine the above relation is reflexive, symmetric and transitive.
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?
Let O be the origin. We define a relation between two points P and Q in a plane if OP = OQ. Show that the relation, so defined is an equivalence relation.
If A = {2, 3, 4}, B = {1, 3, 7} and R = {(x, y) : x ∈ A, y ∈ B and x < y} is a relation from A to B, then write R−1.
Define a transitive relation ?
R is a relation on the set Z of integers and it is given by
(x, y) ∈ R ⇔ | x − y | ≤ 1. Then, R is ______________ .
If A = {a, b, c}, then the relation R = {(b, c)} on A is _______________ .
R is a relation from {11, 12, 13} to {8, 10, 12} defined by y = x − 3. Then, R−1 is ______________ .
Let R = {(a, a), (b, b), (c, c), (a, b)} be a relation on set A = a, b, c. Then, R is _______________ .
Let A = {1, 2, 3}. Then, the number of equivalence relations containing (1, 2) is ______.
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.
Give an example of a map which is one-one but not onto
Give an example of a map which is not one-one but onto
Let us define a relation R in R as aRb if a ≥ b. Then R is ______.
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 ______.
R = {(1, 1), (2, 2), (1, 2), (2, 1), (2, 3)} be a relation on A, then R is ____________.
Let A = {1, 2, 3}. Which of the following is not an equivalence relation on A?
If f(x) = `1 - 1/"x", "then f"("f"(1/"x"))` ____________.
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 ____________.
Let A = {1, 2, 3, …. n} and B = {a, b}. Then the number of surjections from A into B 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}
- Let R be a relation on B defined by R = {(1,2), (2,2), (1,3), (3,4), (3,1), (4,3), (5,5)}. Then R 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 number of surjective functions from A to B where A = {1, 2, 3, 4} and B = {a, b} is
A relation in a set 'A' is known as empty relation:-
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 f(x)= ax2 + bx + c be such that f(1) = 3, f(–2) = λ and f(3) = 4. If f(0) + f(1) + f(–2) + f(3) = 14, then λ is equal to ______.
