Advertisements
Advertisements
प्रश्न
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.
उत्तर
(i) Reflexive
R = {(L1, L2): L1 is parallel to L2}
R is reflexive as any line L1 is parallel to itself, i.e., (L1, L1) ∈ R.
∴ R is reflexive
(ii) Symmetric
Now, let (L1, L2) ∈ R
⇒ L1 is parallel to L2
⇒ L2 is parallel to L1
⇒ (L2, L1) ∈ R
∴ R is symmetric
(iii) Transitive
Now, let (L1, L2), (L2, L3) ∈ R
⇒ L1 is parallel to L2. Also, L2 is parallel to L3
⇒ L1 is parallel to L3
∴R is transitive
Hence, R is an equivalence relation
The set of all lines related to the line y = 2x + 4 is the set of all lines that are parallel to the line y = 2x+ 4
Slope of line y = 2x + 4 is m = 2
It is known that parallel lines have the same slopes
The line parallel to the given line is of the form y = 2x + c, where c ∈ R
Hence, the set of all lines related to the given line is given by y = 2x + c, where c ∈ R
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)].
Let N denote the set of all natural numbers and R be the relation on N × N defined by (a, b) R (c, d) if ad (b + c) = bc (a + d). Show that R is an equivalence relation.
Show that each of the relation R in the set A= {x ∈ Z : 0 ≤ x ≤ = 12} given by R = {(a,b) : a = b} is an equivalence relation. Find the set of all elements related to 1 in each case.
Given an example of a relation. Which is Symmetric but neither reflexive nor transitive.
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.
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 of the following relation is reflexive, symmetric and transitive:
R = {(x, y) : x is father of and y}
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
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.
Give an example of a relation which is symmetric and transitive but not reflexive?
Give an example of a relation which is transitive but neither reflexive nor symmetric?
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?
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.
Let A = {2, 3, 4, 5} and B = {1, 3, 4}. If R is the relation from A to B given by a R b if "a is a divisor of b". Write R as a set of ordered pairs.
For the set A = {1, 2, 3}, define a relation R on the set A as follows:
R = {(1, 1), (2, 2), (3, 3), (1, 3)}
Write the ordered pairs to be added to R to make the smallest equivalence relation.
Write the smallest equivalence relation on the set A = {1, 2, 3} ?
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 _______________ .
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 ______________ .
Show that the relation R on the set Z of all integers, given by R = {(a,b) : 2 divides (a-b)} is an equivalence relation.
In the set of natural numbers N, define a relation R as follows: ∀ n, m ∈ N, nRm if on division by 5 each of the integers n and m leaves the remainder less than 5, i.e. one of the numbers 0, 1, 2, 3 and 4. Show that R is equivalence relation. Also, obtain the pairwise disjoint subsets determined by R
Let n be a fixed positive integer. Define a relation R in Z as follows: ∀ a, b ∈ Z, aRb if and only if a – b is divisible by n. Show that R is an equivalance relation
Give an example of a map which is not one-one but onto
Let R be the relation on N defined as by x + 2 y = 8 The domain of R is ____________.
Let A = {1, 2, 3}. Which of the following is not an equivalence relation on A?
A relation R on a non – empty set A is an equivalence relation if it 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 ____________.
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}
- Raji wants to know the number of relations possible from A to B. How many numbers of relations are possible?
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?
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- Let relation R be defined by R = {(L1, L2): L1║L2 where L1, L2 ∈ L} then R is ____________ relation.
Students of Grade 9, planned to plant saplings along straight lines, parallel to each other to one side of the playground ensuring that they had enough play area. Let us assume that they planted one of the rows of the saplings along the line y = x − 4. Let L be the set of all lines which are parallel on the ground and R be a relation on L.
Answer the following using the above information.
- Let R = {(L1, L2 ): L1 is parallel to L2 and L1: y = x – 4} then which of the following can be taken as L2?
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:-
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 ______.
If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.
