Advertisements
Advertisements
Question
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?
Advertisements
Solution
(i) Reflexive:
R = {(T1, T2) : T1 is similar to T2}
R is reflexive since every triangle is similar to itself.
∴ R is reflexive.
(ii) Symmetric:
Further, if (T1, T2) ∈ R, then T1 is similar to T2.
⇒ T2 is similar to T1.
⇒ (T2, T1) ∈ R
∴ R is symmetric.
(iii) Transitive:
Now, let (T1, T2), (T2, T3) ∈ R.
⇒ T1 is similar to T2 and T2 is similar to T3.
⇒ T1 is similar to T3.
⇒ (T1, T3) ∈ R
∴ R is transitive.
Thus, R is an equivalence relation.
Now, we can observe that:
`3/6 = 4/8 = 5/10 (=1/2)`
The corresponding sides of triangles T1 and T3 are in the same ratio.
Then, triangle T1 is similar to triangle T3.
Hence, T1 is related to T3.
APPEARS IN
RELATED QUESTIONS
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 in the set R of real numbers, defined as R = {(a, b) : a ≤ b2} is neither reflexive nor symmetric nor transitive.
Given an example of a relation. Which is transitive but neither reflexive nor symmetric.
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 1 + ab > 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.
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.
Show that the relation R on the set Z of integers, given by
R = {(a, b) : 2 divides a – b}, is an equivalence relation.
Prove that the relation R on Z defined by
(a, b) ∈ R ⇔ a − b is divisible by 5
is an equivalence relation on Z.
Define an equivalence relation ?
A = {1, 2, 3, 4, 5, 6, 7, 8} and if R = {(x, y) : y is one half of x; x, y ∈ A} is a relation on A, then write R as a set of ordered pairs.
If a relation R is defined on the set Z of integers as follows:
(a, b) ∈ R ⇔ a2 + b2 = 25. Then, domain (R) is ___________
The relation R = {(1, 1), (2, 2), (3, 3)} on the set {1, 2, 3} 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.
If A = {a, b, c}, B = (x , y} find A × A.
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}. Find A × (B ∪ C).
Let A = {6, 8} and B = {1, 3, 5}.
Let R = {(a, b)/a∈ A, b∈ B, a – b is an even number}. Show that R is an empty relation from A to B.
Let A = {0, 1, 2, 3} and define a relation R on A as follows: R = {(0, 0), (0, 1), (0, 3), (1, 0), (1, 1), (2, 2), (3, 0), (3, 3)}. Is R reflexive? symmetric? transitive?
Give an example of a map which is neither one-one nor onto
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 in N by aRb if 2a + 3b = 30. Then R = ______.
Let R = {(3, 1), (1, 3), (3, 3)} be a relation defined on the set A = {1, 2, 3}. Then R is symmetric, transitive but not reflexive.
Let A = {1, 2, 3} and R = {(1, 2), (2, 3), (1, 3)} be a relation on A. Then, R is ____________.
A relation R on a non – empty set A is an equivalence relation if it is ____________.
Let R be the relation “is congruent to” on the set of all triangles in a plane is ____________.
The relation R is defined on the set of natural numbers as {(a, b) : a = 2b}. Then, R-1 is given by ____________.
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?
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:
A relation 'R' in a set 'A' is called reflexive, if
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 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 ______.
Which relation is an example of an Empty Relation for \(A=\{1,2\}\)?
Which statement defines a Universal Relation?
What should be checked to determine whether a relation is transitive?
