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If R is a relation from a non – empty set A to a non – empty set B, then
Let A and B be two sets. Then a relation R from set A to set B is a subset of A × B. Thus, R is a relation from A to B ⇔ R ⊆ A × B.
Here, 0 ≤ x 3 ≤ 7  x
⇒0 ≤ x  3 and x  3 ≤ 7  x
By solvation, we will get 3 ≤ x ≤ 5
So x = 3,4,5 find the values of ^{7x} P_{x  3} by substituting the values of x
Let R be the relation over the set of straight lines of a plane such that l_{1} R l_{2} ⇔ l_{1} ⊥ l_{2}. Then, R is
To be reflexive, a line must be perpendicular to itself, but which is not true. So, R is not reflexive
For symmetric, if l_{1} R l_{2} ⇒ l_{1} ⊥ l_{2}.
⇒ l_{2} ⊥ l_{1} ⇒ l_{1} R l_{2} hence symmetric
For transitive, if l_{1} R l_{2} and l_{2} R l_{3}
⇒ l_{1} R l_{2} and l_{2} R l_{3} does not imply that l_{1} ⊥ l_{3} hence not transitive.
Because, the element b in the domain A has no image in the codomain B.
Because, f( x) = f(x) is the necessary condition for a function to be an even function, which is only satisfied by x^{2}+ sin^{2}x .
The binary relation S = Φ (empty set) on set A = {1, 2, 3} is
Reflexive : A relation is reflexive if every element of set is paired with itself. Here none of the element of A is paired with themselves, so S is not reflexive.
Symmetric : This property says that if there is a pair (a, b) in S, then there must be a pair (b, a) in S. Since there is no pair here in S, this is trivially true, so S is symmetric.
Transitive : This says that if there are pairs (a, b) and (b, c) in S, then there must be pair (a,c) in S. Again, this condition is trivially true, so S is transitive.
The void relation (a subset of A x A) on a non empty set A is:
The relation { } ⊂ A x A on a is surely not reflexive. However, neither symmetry nor transitivity is contradicted. So { } is a transitive and symmetry relation on A.
A relation R on a non empty set A is said to be reflexive if fx Rx for all x ∈ R, Therefore, R is reflexive.
The domain of the function f = {(1, 3), (3, 5), (2, 6)} is
The domain in ordered pair (x,y) is represented by xcoordinate. Therefore, the domain of the given function is given by : {1, 3, 2}.
x  1 ≥ 0 and 6 – x ≥ 0 ⇒ 1 ≤ x ≤ 6.
Let R be the relation on N defined as x R y if x + 2 y = 8. The domain of R is
As x R y if x + 2y = 8, therefore, domain of the relation R is given by x = 8 – 2y ∈ N.
When y = 1,
⇒ x = 6 ,when y = 2,
⇒ x = 4, when y = 3,
⇒ x = 2.
therefore domain is {2, 4, 6}.
If n ≥ 2, then the number of onto mappings or surjections that can be defined from {1, 2, 3, 4, ……….., n} onto {1, 2} is
The number of onto functions that can be defined from a finite set A containing n elements onto a finite set B containing 2 elements = 2^{n }− 2.
A relation R on a non empty set A is said to be symmetric if fx Ry ⇔ yRx, for all x , y ∈ R .
We have ,
Therefore, range of f(x) is {1}.
For even function: f(x) = f(x) ,
therefore, f(− x)
= sin (− x)^{2} = sin x^{2} = f(x).
Which of the following is not an equivalence relation on I, the set of integers ; x, y
If R is a relation defined by xRy : ifx ⩽ y, then R is reflexive and transitive But, it is not symmetric. Hence, R is not an equivalence relation.
If A = {1, 2, 3}, then the relation R = {(1, 2), (2, 3), (1, 3) in A is
A relation R on a non empty set A is said to be transitive if fxRy and y Rz ⇒ xRz, for all x ∈ R. Here, (1, 2) and (2, 3) belongs to R implies that (1, 3) belongs to R.
A relation R on a non empty set A is said to be transitive if fx Ry and yRz ⇒ x Rz, for all x ∈ R.
We have, f(x) = x−1, which always gives nonnegative values of f(x) for all x ∈ R.Therefore range of the given function is all nonnegative real numbers i.e. [0,∞).
As the denominator of the function is a modulus function i.e.
Let A = {a, b, c} and R = {(a, a), (b, b), (c, c), (b, c)} be a relation on A. Here, R is
Correct Answer : b
Explanation: A = {a, b, c} and R = {(a, a), (b, b), (c, c), (b, c)}
Any relation R is reflexive if fx Rx for all x ∈ R. Here ,(a, a), (b, b), (c, c) ∈ R. Therefore , R is reflexive.
For the transitive, in the relation R there should be (a,c)
Hence it is not transitive.
A relation R from C to R is defined by x Ry iff x = y. Which of the following is correct?
A relation R in a set A is said to be an equivalence relation if
A relation R on a non empty set A is said to be reflexive iff xRx for all x ∈ R . .
A relation R on a non empty set A is said to be symmetric if fx Ry ⇔ y Rx, for all x , y ∈ R .
A relation R on a non empty set A is said to be transitive if fx Ry and y Rz ⇒ x Rz, for all x ∈ R.
An equivalence relation satisfies all these three properties.
Let f: R → R be a mapping such that f(x) = . Then f is
A polynomial function has all exponents as integral whole numbers.
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