Sets and Relations

Let $A_1, A_2, \ldots, A_6$ are six sets, each with four elements and $B_1, B_2, \ldots ., B_n$ are $n$ sets, each with two elements. Let $S=A_1 \cup A_2 \cup \ldots \cup A_6=B_1 \cup B_2 \cup \ldots \cup B_n$.
Given that each element of $S...

On the set $\mathbb{R}$ of real numbers the relation $\rho$, defined by $\mathrm{x} \rho \mathrm{y}(\mathrm{x}, \mathrm{y} \in \mathbb{R})$ iff

The number of reflexive relations on a set $A$ of $n$ elements is equal to

For the real numbers \(x\) & \(y\), we write \(x\) p y iff \(x-y+\sqrt{2}\) is an irrational number. Then relation p is

Let A be the set of even natural numbers that are < 8 & B be the set of prime integers that are \(<7\) The number of relations from A to B are

In R, a relation p is defined as follows:
\(\forall a, b \in \mathbb{R}, a p\) holds iff \(a^2-4 a b+3 b^2=0\). Then

If R and R\(^1\) are equivalence relations on a set A, then so are the relations

Let \(\rho\) be a relation defined on set of natural numbers N, as \(\rho = \{ (x,y) \in N \times N:2x + y = 4\}\). Then domain A and range B are

Let X be a nonvoid set. If \(\rho_1\) and \(\rho_2\) be the transitive relations on X, then
(\(\circ\) denotes the composition of relations)

Let A, B, C are subsets of set X. Then consider the validity of the following set theoretic statement:

Let R and S be two equivalence relations on a non-void set A. Then

For the mapping \(f:R - \{ 1\} \to R - \{ 2\}\), given by \(f(x) = {{2x} \over {x - 1}}\), which of the following is correct?

Let S, T, U be three non-void sets and f : S \(\to\) T, g : T \(\to\) U and composed mapping g . f : S \(\to\) U be defined. Let g . f be injective mapping. Then

A is a set containing n elements. P and Q are two subsets of A. Then the number of ways of choosing P and Q so that P \(\cap\) Q = \(\varphi\) is

Let R be the real line. Let the relations S and T or R be defined by \(S = \{ (x,y):y = x + 1,0 < x < 2\} ,T = \{ (x,y):x - y\) is an integer}. Then

Let A, B, C be three non-void subsets of set S. Let (A \(\cap\) C) \(\cup\) (B \(\cap\) C') = \(\phi\) where C' denote the complement of set C in S. Then

Let p1 and p2 be two equivalence relations defined on a non-void set S. Then

Let the relation p be defined on R by a p b holds if and only if a \(-\) b is zero or irrational, then

Let the relation \(\rho\) be defined on R as a\(\rho\)b if 1 + ab > 0. Then,

Let S, T, U be three non-void sets and f : S \(\to\) T, g : T \(\to\) U be so that gof : s \(\to\) U is surjective. Then,

Let f : X \(\to\) Y and A, B are non-void subsets of Y, then (where the symbols have their usual interpretation)

On the set R of real numbers, the relation \(\rho\) is defined by x\(\rho\)y, (x, y) \(\in\) R.

Let \(\rho\) be a relation defined on N, the set of natural numbers, as\(\rho\) = {(x, y) \(\in\) N \(\times\) N : 2x + y = 41}. Then

On R, a relation \(\rho\) is defined by x\(\rho\)y if and only if x \(-\) y is zero or irrational. Then,

On R, the set of real numbers, a relation \(\rho\) is defined as 'a\(\rho\)b if and only if 1 + ab > 0'. Then,

On the set R of real numbers we define xPy if and only if xy \(\ge\) 0. Then, the relation P is

On set A = {1, 2, 3}, relations R and S are given byR = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 1)},S = {(1, 1), (2, 2), (3, 3), (1, 3), (3, 1)}. Then,

On R, the relation \(\rho\) be defined by 'x\(\rho\)y holds if and only if x \(-\) y is zero or irrational'. Then,

If A = {5n \(-\) 4n \(-\) 1 : n\(\in\)N} and B = {16(n \(-\) 1) : n\(\in\)N}, then

Let R be a relation defined on the set Z of all integers and xRy, when x + 2y is divisible by 3, then

The function \(f(x) = \sec \left[ {\log \left( {x + \sqrt {1 + {x^2}} } \right)} \right]\) is

If A = {1, 2, 3, 4}, B = {1, 2, 3, 4, 5, 6} are two sets, and function f : A \(\to\) B is defined by f(x) = x + 2 \(\forall\) x\(\in\) A, then the function f is

Let R be the set of real numbers and the mapping f : R \(\to\) R and g : R \(\to\) R be defined by f(x) = 5 \(-\) x2 and g(x) = 3x \(-\) 4, then the value of (fog)(\(-\)1) is

For any two sets A and B, A \(-\) (A \(-\) B) equals

Let A = {1, 2, 3} and B = {2, 3, 4}, then which of the following relations is a function from A to B?

A function f : A \(\to\) B, where A = {x/\(-\)1 \(\le\) x \(\le\) 1} and B = {y/1 \(\le\) y \(\le\) 2} is defined by the rule y = f(x) = 1 + x2. Which of the following statement is then true?

The mapping f : N \(\to\) N given by f(n) = 1 + n2, n \(\in\) N where N is the set of natural numbers, is

A mapping f : N \(\to\) N where N is the set of natural numbers is defined as
f(n) = n2 for n odd
f(n) = 2n + 1 for n even
for n \(\in\) N. Then f is

Three sets A, B, C are such that A = B \(\cap\) C and B = C \(\cap\) A, then