English

Given an example of a relation. Which is symmetric and transitive but not reflexive.

Advertisements
Advertisements

Question

Given an example of a relation. Which is symmetric and transitive but not reflexive.

Sum
Advertisements

Solution

Let A = {1, 2}.

Define a relation R on A as:

R = {(1, 1)}

R is symmetric, because if (1, 1) ∈ R, then (1, 1) ∈ R.

R is transitive, because (1, 1) and (1, 1) imply (1, 1) ∈ R.

R is not reflexive, because (2, 2) ∉ R.

Hence, R is symmetric and transitive but not reflexive.

shaalaa.com
  Is there an error in this question or solution?
Chapter 1: Relations and Functions - EXERCISE 1.1 [Page 6]

APPEARS IN

NCERT Mathematics Part 1 and 2 [English] Class 12
Chapter 1 Relations and Functions
EXERCISE 1.1 | Q 10. (v) | Page 6

RELATED QUESTIONS

Check whether the relation R in R defined by R = {(a, b) : a ≤ b3} is reflexive, symmetric or 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 of all the books in a library of a college, given by R = {(x, y) : x and y have the same number of pages} is an equivalence relation.


Test whether the following relation R1 is  (i) reflexive (ii) symmetric and (iii) transitive :

R1 on Q0 defined by (a, b) ∈ R1 ⇔ = 1/b.


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


Prove that every identity relation on a set is reflexive, but the converse is not necessarily true.


Show that the relation '≥' on the set R of all real numbers is reflexive and transitive but not symmetric ?


Give an example of a relation which is reflexive and transitive but not symmetric?


Let A = {abc} and the relation R be defined on A as follows: R = {(aa), (bc), (ab)}. Then, write minimum number of ordered pairs to be added in R to make it reflexive and transitive.


Defines a relation on N:

xy is square of an integer, x, y ∈ N

Determine the above relation is reflexive, symmetric and transitive.


Show that the relation R defined by R = {(a, b) : a – b is divisible by 3; a, b ∈ Z} 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.


Write the domain of the relation R defined on the set Z of integers as follows:-
(a, b) ∈ R ⇔ a2 + b2 = 25


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.


Let the relation R be defined on N by aRb iff 2a + 3b = 30. Then 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 ______________ .


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 _____________ .


If A = {a, b, c}, then the relation R = {(b, c)} on A is _______________ .


Let A = {2, 3, 4, 5, ..., 17, 18}. Let '≃' be the equivalence relation on A × A, cartesian product of Awith itself, defined by (a, b) ≃ (c, d) if ad = bc. Then, the number of ordered pairs of the equivalence class of (3, 2) is _______________ .


Let R be a relation on N defined by x + 2y = 8. The domain of R is _______________ .


Let R = {(a, a), (b, b), (c, c), (a, b)} be a relation on set A = a, b, c. Then, R is _______________ .


If R is a relation on the set A = {1, 2, 3, 4, 5, 6, 7, 8, 9} given by x R y ⇔ y = 3 x, then R = _____________ .


If R is a relation on the set A = {1, 2, 3} given by R = {(1, 1), (2, 2), (3, 3)}, then R is ____________ .


Let R be the relation on the set A = {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Then, _____________________ .


Mark the correct alternative in the following question:

For real numbers x and y, define xRy if `x-y+sqrt2` is an irrational number. Then the relation R is ___________ .


Show that the relation R on R defined as R = {(a, b): a ≤ b}, is reflexive, and transitive but not symmetric.


Show that the relation S in the set A = [x ∈ Z : 0 ≤ x ≤ 12] given by S = [(a, b) : a, b ∈ Z, ∣a − b∣ is divisible by 3] is an equivalence relation.


If A = {a, b, c}, B = (x , y} find A × A.


Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of the following:
an injective mapping from A to B


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 A = {1, 2, 3, …. n} and B = {a, b}. Then the number of surjections from A into B is ____________.


Let S = {1, 2, 3, 4, 5} and let A = S x S. Define the relation R on A as follows:
(a, b) R (c, d) iff ad = cb. Then, R is ____________.


The value of k for which the system of equations x + ky + 3z = 0, 4x + 3y + kz = 0, 2x + y + 2z = 0 has nontrivial solution is


On the set N of all natural numbers, define the relation R by a R b, if GCD of a and b is 2. Then, R is


A relation 'R' in a set 'A' is called reflexive, if


If f(x + 2a) = f(x – 2a), then f(x) is:


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.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×