NCERT solutions for माठेमटिक्स पार्ट १ अँड २ [इंग्रजी] इयत्ता १२ chapter 1 - Relations and Functions [Latest edition]

Determine whether the following relation is reflexive, symmetric and transitive:

Relation R in the set A = {1, 2, 3, ..., 13, 14} defined as R = {(x, y) : 3x − y = 0}.

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 A = {1, 2, 3, 4, 5, 6} as R = {(x, y) : y is divisible by x}.

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}.

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}

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 live in the same locality}

Determine whether the following relation is reflexive, symmetric and transitive:

Relation R in the set A of human beings in a town at a particular time given by R = {(x, y) : x is exactly 7 cm taller than y}.

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 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 father of and y}

Show that the relation R in the set R of real numbers, defined as R = {(a, b) : a ≤ b²} is neither reflexive nor symmetric nor transitive.

Check whether the relation R defined in the set {1, 2, 3, 4, 5, 6} as R = {(a, b) : b = a + 1} is reflexive, symmetric, or transitive.

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

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.

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 a multiple of 4} is an equivalence relation. Find the set of all elements related to 1.

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.

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

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

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

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

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

Show that the relation R in the set A of points in a plane given by R = {(P, Q) : distance of the point P from the origin is the same as the distance of the point Q from the origin} is an equivalence relation. Further, show that the set of all points related to a point P ≠ (0, 0) is the circle passing through P with the origin as its centre.

Show that the relation R defined in the set A of all triangles as R = {(T₁, T₂) : T₁ is similar to T₂}, is an equivalence relation. Consider three right angle triangles T₁ with sides 3, 4, 5, T₂ with sides 5, 12, 13 and T₃ with sides 6, 8, and 10. Which triangles among T₁, T₂ and T₃ are related?

Show that the relation R, defined in the set A of all polygons as R = {(P₁, P₂) : P₁ and P₂ have the same number of sides}, is an equivalence relation. What is the set of all elements in A related to the right-angled triangle T with sides 3, 4 and 5?

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 R be the relation in the set {1, 2, 3, 4} given by R = {(1, 2), (2, 2), (1, 1), (4, 4), (1, 3), (3, 3), (3, 2)}. Choose the correct answer.

Let R be the relation in the set N given by R = {(a, b) : a = b − 2, b > 6}. Choose the correct answer.

Show that the function f : R_* → R_* defined by f(x) = `1/x` is one-one and onto, where R_* is the set of all non-zero real numbers. Is the result true if the domain R_* is replaced by N, with the co-domain being the same as R?

Check the injectivity and surjectivity of the following function:

f : N → N given by f(x) = x²

Check the injectivity and surjectivity of the following function:

f : Z → Z given by f(x) = x²

Check the injectivity and surjectivity of the following function:

f : R → R given by f(x) = x²

Check the injectivity and surjectivity of the following function:

f : N → N given by f(x) = x³

Check the injectivity and surjectivity of the following function:

f : Z → Z given by f(x) = x³

Prove that the greatest integer function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.

Show that the modulus function f : R → R given by f(x) = |x| is neither one-one nor onto, where |x| is x if x is positive or 0 and |x| is − x if x is negative.

Show that the Signum Function f : R → R, given by `f(x) = {(1", if"  x > 0), (0", if"  x  = 0), (-1", if"  x < 0):}` is neither one-one nor onto.

Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.

In the following case, state whether the function is one-one, onto or bijective. Justify your answer.

f : R → R defined by f(x) = 3 − 4x

In the following case, state whether the function is one-one, onto or bijective. Justify your answer.

f : R → R defined by f(x) = 1 + x²

Let A and B be sets. Show that f : A × B → B × A such that f(a, b) = (b, a) is a bijective function.

Let f : N → N be defined by f(n) = `{((n+1)/2", if n is odd"),(n/2", if n is even"):}` for all n ∈ N.

State whether the function f is bijective. Justify your answer.

Let A = R − {3} and B = R − {1}. Consider the function f : A → B defined by f(x) = `((x- 2)/(x -3))`. Is f one-one and onto? Justify your answer.

Let f : R → R be defined as f(x) = x⁴. Choose the correct answer.

Let f : R → R be defined as f(x) = 3x. Choose the correct answer.

Show that the function f : R → {x ∈ R : −1 < x < 1} defined by f(x) = `x/(1 + |x|)`, x ∈ R is one-one and onto function.

Show that the function f : R → R given by f(x) = x³ is injective.

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.

Find the number of all onto functions from the set {1, 2, 3, ..., n} to itself.

Let A = {−1, 0, 1, 2}, B = {−4, −2, 0, 2} and f, g : A → B be functions defined by f(x) = x² − x, x ∈ A and g(x) = `2|x - 1/2|- 1`, x ∈ A. Are f and g equal?

Justify your answer. (Hint: One may note that two functions f : A → B and g : A → B such that f(a) = g(a) ∀ a ∈ A are called equal functions.)

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 ______.

Let A = {1, 2, 3}. Then, the number of equivalence relations containing (1, 2) is ______.