मराठी

Define a Symmetric Relation ?

Advertisements
Advertisements

प्रश्न

Define a symmetric relation ?

व्याख्या
Advertisements

उत्तर

A relation R on a set A is said to be symmetric if

(a, b) ∈ R

⇒  (b, a)  ∈ R for all b ∈ A

i.e.  aR⇒ bRa for all , ∈ A

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 1: Relations - Exercise 1.3 [पृष्ठ ३०]

APPEARS IN

आर.डी. शर्मा Mathematics Volume 1 and 2 [English] Class 12
पाठ 1 Relations
Exercise 1.3 | Q 11 | पृष्ठ ३०

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}.


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}.


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 A = {x ∈ Z : 0 ≤ x ≤ 12} given by R = {(a, b) : |a – b| is a multiple of 4} is an equivalence relation. Find the set of all elements related to 1.


Given a non-empty set X, consider P(X), which is the set of all subsets of X. Define the relation R in P(X) as follows:

For subsets A, B in P(X), ARB if and only if A ⊂ B. Is R an equivalence relation on P(X)? Justify your answer.


Let A = {x ∈ Z : 0 ≤ x ≤ 12}. Show that R = {(ab) : a∈ A, |a – b| is divisible by 4}is an equivalence relation. Find the set of all elements related to 1. Also write the equivalence class [2]


The binary operation *: R x R → R is defined as a *b = 2a + b Find (2 * 3)*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 wife of y}


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.


If = {1, 2, 3, 4} define relations on A which have properties of being reflexive, symmetric and transitive ?


Give an example of a relation which is symmetric but neither reflexive nor transitive?


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?


Write the smallest reflexive relation on set A = {1, 2, 3, 4}.


State the reason for the relation R on the set {1, 2, 3} given by R = {(1, 2), (2, 1)} to be transitive ?


If a relation R is defined on the set Z of integers as follows:
(a, b) ∈ R ⇔ a2 + b2 = 25. Then, domain (R) is ___________


Let A = {1, 2, 3}. Then, the number of relations containing (1, 2) and (1, 3) which are reflexive and symmetric but not transitive is ______.


If R is the largest equivalence relation on a set A and S is any relation on A, then _____________ .


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


The relation R = {(1, 1), (2, 2), (3, 3)} on the set {1, 2, 3} is ___________________ .


Mark the correct alternative in the following question:

Let L denote the set of all straight lines in a plane. Let a relation R be defined by lRm if l is perpendicular to m for all l, m  L. Then, R is ______________ .


Mark the correct alternative in the following question:

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 for all a, b  T. Then, R is ____________ .


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


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


Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6} Find (A × B) ∩ (A × C).


Write the relation in the Roster form and hence find its domain and range :
R1 = {(a, a2) / a is prime number less than 15}


Write the relation in the Roster form and hence find its domain and range:

R2 = `{("a", 1/"a")  "/"  0 < "a" ≤ 5, "a" ∈ "N"}`


The following defines a relation on N:
x + 4y = 10 x, y ∈ N.
Determine which of the above relations are reflexive, symmetric and transitive.


The maximum number of equivalence relations on the set A = {1, 2, 3} are ______.


If a relation R on the set {1, 2, 3} be defined by R = {(1, 2)}, then R is ______.


Let A = {1, 2, 3}. Which of the following is not an equivalence relation on A?


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 ____________.


A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever


Let I be the set of all citizens of India who were eligible to exercise their voting right in the general election held in 2019. A relation ‘R’ is defined on I as follows:

R = {(V1, V2) ∶ V1, V2 ∈ I and both use their voting right in the general election - 2019}

  • Mr. Shyam exercised his voting right in General Election-2019, then Mr. Shyam is related to which of the following?

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: B → B be defined by R = {(1,1),(1,2), (2,2), (3,3), (4,4), (5,5), (6,6)}, then R is ____________.

A relation 'R' in a set 'A' is called a universal relation, if each element of' A' is related to :-


Given a non-empty set X, define the relation R in P(X) as follows:

For A, B ∈ P(X), (4, B) ∈ R iff A ⊂ B. Prove that R is reflexive, transitive and not symmetric.


Read the following passage:

An organization conducted bike race under two different categories – Boys and Girls. There were 28 participants in all. 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} and G = {g1, g2}, where B represents the set of Boys selected and G the set of Girls selected for the final race.

Based on the above information, answer the following questions:

  1. How many relations are possible from B to G? (1)
  2. Among all the possible relations from B to G, how many functions can be formed from B to G? (1)
  3. Let R : B `rightarrow` B be defined by R = {(x, y) : x and y are students of the same sex}. Check if R is an equivalence relation. (2)
    OR
    A function f : B `rightarrow` G be defined by f = {(b1, g1), (b2, g2), (b3, g1)}. Check if f is bijective. Justify your answer. (2)

Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×