Advertisements
Advertisements
प्रश्न
Show that the relation R defined in the set A of all polygons as R = {(P1, P2): P1 and P2have same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right angle triangle T with sides 3, 4 and 5?
Advertisements
उत्तर
R = {(P1, P2): P1 and P2 have same the number of sides}
R is reflexive since (P1, P1) ∈ R as the same polygon has the same number of sides with itself.
Let (P1, P2) ∈ R.
⇒ P1 and P2 have the same number of sides.
⇒ P2 and P1 have the same number of sides.
⇒ (P2, P1) ∈ R
∴R is symmetric.
Now,
Let (P1, P2), (P2, P3) ∈ R.
⇒ P1 and P2 have the same number of sides. Also, P2 and P3 have the same number of sides.
⇒ P1 and P3 have the same number of sides.
⇒ (P1, P3) ∈ R
∴R is transitive.
Hence, R is an equivalence relation.
The elements in A related to the right-angled triangle (T) with sides 3, 4, and 5 are those polygons which have 3 sides (since T is a polygon with 3 sides).
Hence, the set of all elements in A related to triangle T is the set of all triangles.
APPEARS IN
संबंधित प्रश्न
Show that the relation R in the set A = {1, 2, 3, 4, 5} given by R = {(a, b) : |a − b| is even}, is an equivalence relation. Show that all the elements of {1, 3, 5} are related to each other and all the elements of {2, 4} are related to each other. But no element of {1, 3, 5} is related to any element of {2, 4}.
Show that 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.
Given an example of a relation. Which is Transitive but neither reflexive nor symmetric.
Given an example of a relation. Which is Reflexive and symmetric but not transitive.
Let L be the set of all lines in the 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.
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 and y work at the same place}
Three relations R1, R2 and R3 are defined on a set A = {a, b, c} as follows:
R1 = {(a, a), (a, b), (a, c), (b, b), (b, c), (c, a), (c, b), (c, c)}
R2 = {(a, a)}
R3 = {(b, c)}
R4 = {(a, b), (b, c), (c, a)}.
Find whether or not each of the relations R1, R2, R3, R4 on A is (i) reflexive (ii) symmetric and (iii) transitive.
Test whether the following relation R3 is (i) reflexive (ii) symmetric and (iii) transitive:
R3 on R is defined by (a, b) ∈ R3 `⇔` a2 – 4ab + 3b2 = 0.
Give an example of a relation which is symmetric but neither reflexive nor transitive?
Let A = {a, b, c} and the relation R be defined on A as follows: R = {(a, a), (b, c), (a, b)}. Then, write minimum number of ordered pairs to be added in R to make it reflexive and transitive.
Let n be a fixed positive integer. Define a relation R on Z as follows:
(a, b) ∈ R ⇔ a − b is divisible by n.
Show that R is an equivalence relation on Z.
Show that the relation R on 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.
Let S be a relation on the set R of all real numbers defined by
S = {(a, b) ∈ R × R : a2 + b2 = 1}
Prove that S is not an equivalence relation on R.
Let Z be the set of all integers and Z0 be the set of all non-zero integers. Let a relation R on Z × Z0be defined as (a, b) R (c, d) ⇔ ad = bc for all (a, b), (c, d) ∈ Z × Z0,
Prove that R is an equivalence relation on Z × Z0.
If R is a symmetric relation on a set A, then write a relation between R and R−1.
If A = {3, 5, 7} and B = {2, 4, 9} and R is a relation given by "is less than", write R as a set ordered pairs.
Let R be the equivalence relation on the set Z of the integers given by R = { (a, b) : 2 divides a - b }.
Write the equivalence class [0].
Let the relation R be defined on the set A = {1, 2, 3, 4, 5} by R = {(a, b) : | a2- b2 | < 8}. Write R as a set of ordered pairs.
If R is the largest equivalence relation on a set A and S is any relation on A, then _____________ .
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, _____________________ .
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.
Let A = {1, 2, 3, 4}, B = {4, 5, 6}, C = {5, 6}. Find A × (B ∩ C).
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.
Write the relation in the Roster form and hence find its domain and range:
R2 = `{("a", 1/"a") "/" 0 < "a" ≤ 5, "a" ∈ "N"}`
Let R be a relation on the set N of natural numbers defined by nRm if n divides m. Then R is ______.
Give an example of a map which is neither one-one nor onto
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 = { 2, 3, 6 } Which of the following relations on A are reflexive?
R = {(1, 1), (2, 2), (1, 2), (2, 1), (2, 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 ____________.
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}
- 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?
If A = {1,2,3}, B = {4,6,9} and R is a relation from A to B defined by ‘x is smaller than y’. The range of R 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
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.
Let A = {3, 5}. Then number of reflexive relations on A is ______.
If a relation R on the set {a, b, c} defined by R = {(b, b)}, then classify the relation.
