Sets, Relations and Functions

Let $\displaystyle A = \{a, b, c, d, e\}$ then the number of proper subsets is

The numbers of proper subset of the set $\displaystyle (3, 4, 5, 6, 7)$ is

If $\displaystyle A = (1, 2, 3, 4, 5, 6, 7)$ and $\displaystyle B = (2, 4, 6, 8)$. Cardinal number of $\displaystyle A - B$ is:

$\displaystyle (A^c)^c = ?$ Note: This que is from matrix (deleted topic).

Two finite sets respectively have $\displaystyle x$ and $\displaystyle y$ number of elements. The total number of subsets of the first is $\displaystyle 56$ more than the total number of subsets of the second. The value of $\displaystyle x$ and $\displaystyle y$ respectively.

The set of cubes of the natural number is:

The number of integers from $\displaystyle 1$ to $\displaystyle 100$ which are neither divisible by $\displaystyle 3$ nor by $\displaystyle 5$ nor by $\displaystyle 7$ is

In a town of $\displaystyle 20,000$ families it was found that $\displaystyle 40\%$ families buy newspaper $\displaystyle A$, $\displaystyle 20\%$ families buy newspaper $\displaystyle B$ and $\displaystyle 10\%$ families buy newspaper $\displaystyle C$, $\displaystyle 5\%$ families buy $\displaystyle A$ and $\displaystyle B$, $\displaystyle 3\%$ buy $\displaystyle B$ and $\displaystyle C$ and $\displaystyle 4\%$ buy $\displaystyle A$ and $\displaystyle C$, if $\displaystyle 2\%$ families buy all the three newspaper, then the number of families which buy only $\displaystyle A$ is:

The number of items in the set $\displaystyle A$ is $\displaystyle 40$; in the set $\displaystyle B$ is $\displaystyle 32$; in the set $\displaystyle C$ is $\displaystyle 50$; in both $\displaystyle A$ and $\displaystyle B$ is $\displaystyle 4$, in both $\displaystyle A$ and $\displaystyle C$ is $\displaystyle 5$; in both $\displaystyle B$ and $\displaystyle C$ is $\displaystyle 7$ in all the sets $\displaystyle 2$. How many are in at least one of the set?

Out of a group of $\displaystyle 20$ teachers in a school, $\displaystyle 10$ teach Mathematics, $\displaystyle 9$ teach Physics and $\displaystyle 7$ teach Chemistry. $\displaystyle 4$ teach Mathematics and Physics but none teach both Mathematics and Chemistry. How many teach Chemistry and Physics; how many teach only Physics?

If $\displaystyle A = \{1, 2, 3, 4, 5, 7, 8, 9\}$ and $\displaystyle B = \{2, 4, 6, 7, 9\}$ then how many proper subset of $\displaystyle A \cap B$ can be created?

The number of subsets of the set $\displaystyle \{0, 1, 2, 3\}$ is:

A survey shows that $\displaystyle 74\%$ of the Canadian like grapes, whereas $\displaystyle 68\%$ like bananas. What percentage of the Canadians like both grapes and bananas, if everybody like either of two?

If $\displaystyle A = \{a, b, c\}$, $\displaystyle B = \{b, c, d\}$ and $\displaystyle C = \{a, d\}$ then $\displaystyle (A - B) \times (B \cap C)$ is equal to:

In a survey of $\displaystyle 100$ boys it was found that $\displaystyle 50$ used white shirts, $\displaystyle 40$ red shirts and $\displaystyle 30$ blue shirts. $\displaystyle 20$ were habituated in using both white and red shirts. $\displaystyle 15$ were using both red and blue shirts and $\displaystyle 10$ were using blue and white shirts. Find the number of boys who are using all colours.

If $\displaystyle A = \{1, 2, 4\}$ and $\displaystyle B = \{1, 2, 3\}$ then $\displaystyle (A \cup B) \times (A \cap B)$ is equal to:

If $\displaystyle B = \{1, 2, 3, 4, 5\}$, then the number of proper subsets of $\displaystyle B$ is

If $\displaystyle A = (1, 2, 3, 4, 5, 6, 7, 8, 9)$ $\displaystyle B = (1, 3, 4, 5, 7, 8)$ $\displaystyle C = (2, 6, 8)$ then find $\displaystyle (A - B) \cup C$

The no. of subsets of the set $\displaystyle (3, 4, 5)$ is:

The set of cubes of natural number is

Two finite sets have $\displaystyle x$ and $\displaystyle y$ number of elements. The total number of subsets of first is $\displaystyle 56$ more than the total number of subsets of second. The value of $\displaystyle x$ and $\displaystyle y$ is:

Given $\displaystyle A = (2, 3)$, $\displaystyle B = (4, 5)$, $\displaystyle C = (5, 6)$ then $\displaystyle A \times (B \cap C)$ is

If the universal set $\displaystyle E = \{x \mid x \text{ is a positive integer } < 25\}$, $\displaystyle A = \{2, 6, 8, 14, 22\}$, $\displaystyle B = \{4, 8, 10, 14\}$

If $\displaystyle A = \{1, 2\}$, $\displaystyle B = \{3, 4\}$, $\displaystyle C = \{5, 6\}$ then the value of $\displaystyle A \times (B \cup C)$

If a set contain $\displaystyle n$ elements, then the total number of proper subsets of set is:

A team has a total population of $\displaystyle 50,000$. Out of it $\displaystyle 28,000$ read the newspaper 'X' and $\displaystyle 23,000$ read newspaper 'Y', while $\displaystyle 4,000$ read both the newspaper. The number of persons not reading any of the two newspapers are:

The number of proper subsets of the set $\displaystyle {3, 4, 5, 6, 7}$ is

If $\displaystyle A = \{1, 2, 3, 4\}$ and $\displaystyle B = \{2, 3, 4, 5\}$ then $\displaystyle (A - B) \cup (B - A)$ is

The number of subsets $\displaystyle {1, 2, 5}$ is

If $\displaystyle A = \{1, 2, 3, 4, 5, 6, 7\}$ and $\displaystyle B = \{2, 4, 6\}$ Cardinal number of $\displaystyle A \cup B$

If $\displaystyle A = \{1, 2, 3, 4\}$ and $\displaystyle B = \{6, 7, 8\}$, then cardinal number of $\displaystyle A \times B$ is:

The number of subsets of the set $\displaystyle A = \{1, 2, 3, 4, 5, 6, 7, 8\}$ is

$\displaystyle (A \cup B)'$ is equal to

If $\displaystyle A = \{p, q, r, s\}$, $\displaystyle B = \{q, s, t\}$ and $\displaystyle C = \{m, q, n\}$ find $\displaystyle C - (A \cap B)$

The set having no element is called

Let $\displaystyle Z$ be the universal set for two sets $\displaystyle A$ and $\displaystyle B$. If $\displaystyle n(A) = 300$, $\displaystyle n(B) = 400$ and $\displaystyle n(A \cup B) = 200$, then $\displaystyle n(A' \cap B')$ is equal to 400 provided $\displaystyle n(Z)$ is equal to

A town has a total population of 50,000. Out of it 28,000 read the newspaper X and 23,000 read Y while 4,000 read both the papers. The number of persons not reading X and Y both is

In a group of students 80 can speak Hindi, 60 can speak English and 40 can speak Hindi and English both, then number of students is:

If $\displaystyle A = \{1, 2, 3, 4, 5\}$ and $\displaystyle B = \{6, 7, 8, 9\}$, then cardinal number of $\displaystyle A \times B$ is:

If $\displaystyle A = \{4, 5\}$, $\displaystyle B = \{2, 3\}$, $\displaystyle C = \{5, 6\}$ then $\displaystyle A \times (B \cap C)$

If $\displaystyle A = \{1, 2, 3\}$, $\displaystyle B = \{3, 4\}$ and $\displaystyle C = \{4, 5, 6\}$, then $\displaystyle A \times (B \cap C)$

Two Finite sets have $\displaystyle m$ and $\displaystyle n$ elements. The total number of subsets of first set is 56 more than the total number of subsets of the second set. The value of $\displaystyle m$ and $\displaystyle n$ are

Let $\displaystyle A$ be the set of squares of natural numbers and let $\displaystyle x \in A, y \in A$, then

In a survey of 300 companies, the number of companies using different Media-Newspapers N(N), Radio (R) and Television (T) are as follows: $\displaystyle N(N) = 200$, $\displaystyle N(R) = 100$, $\displaystyle N(T) = 40$, $\displaystyle N(N \cap R) = 20$, $\displaystyle N(R \cap T) = 20$, $\displaystyle N(N \cap T) = 30$ and $\displaystyle N(N \cap R \cap T) = 5$. Find the numbers of companies using none of these media:

Out of total 150 students, 45 passed in Accounts, 30 in Economics and 50 in Maths, 30 in both Accounts and Maths, 32 in both Maths and Economics, 35 in both Accounts and Economics, 25 students passed in all the three subjects. Find the numbers who passed at least in any one of the subjects:

If $\displaystyle A = \{0, 1, 2, 3, 4, 5\}$ then the no. of subsets of $\displaystyle A$

The number of proper subsets of $\displaystyle A \cap B$, if $\displaystyle A = \{1, 2, 3, 4, 5, 7, 8, 9, 10\}$ and $\displaystyle B = \{2, 4, 6, 7, 9\}$

Out of 20 members in a family, 11 like to take tea and 14 like coffee. Assume that each one likes at least one of the two drinks. Find how many like both coffee and tea:

In a survey of 300 companies, the number of companies using different media-Newspapers (N), Radio (R) and Television (T) are as follows: $\displaystyle N(N) = 200$, $\displaystyle N(R) = 100$, $\displaystyle N(T) = 40$, $\displaystyle N(N \cap R) = 20$, $\displaystyle N(R \cap T) = 20$, $\displaystyle N(N \cap T) = 30$ and $\displaystyle N(N \cap R \cap T) = 5$. Find the numbers of companies using none of these media:

If $\displaystyle A = \{p, q, r, s\}$, $\displaystyle B = \{q, s, t\}$, $\displaystyle C = \{m, q, n\}$ Find $\displaystyle C - [A \cap B]$

From a group of 200 persons, 100 are interested in music, 70 in photography and 40 in swimming, furthermore 40 are interested in both music and photography, 30 in both music and swimming, 20 in photography and swimming and 10 in all the three. How many are interested in photography but not in music and swimming?

If $\displaystyle A = \{a, b, p\}$, $\displaystyle B = \{2, 3\}$, $\displaystyle C = \{p, q, r, s\}$ then $\displaystyle n[(A \cup C) \times B]$ is :

If $\displaystyle A = \{1, 2\}$ and $\displaystyle B = \{3, 4\}$, Determine the number of relations from $\displaystyle A$ and $\displaystyle B$:

In the set of all straight lines on a plane which of the following is not "TRUE"?

If $\displaystyle A$ is related to $\displaystyle B$ if and only if the difference in $\displaystyle A$ and $\displaystyle B$ is an even integer. This relation is

Let $\displaystyle A = \{1, 2, 3\}$ and consider the relation $\displaystyle R = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\}$ then $\displaystyle R$ is:

Given the relation $\displaystyle R = \{(1,2), (2,3)\}$ on the set $\displaystyle A = \{1,2,3\}$, the minimum number of ordered pairs which when added to $\displaystyle R$ make it equivalence relation is

If $\displaystyle R$ be a relation defined on the set of Natural numbers as "$\displaystyle x$ $\displaystyle R$ $\displaystyle y \Leftrightarrow (x-y)$ is divisible by $\displaystyle 5$" $\displaystyle \forall x, y \in N$ then the relation $\displaystyle R$ is

Let $\displaystyle A = \{1,2,3\}$ and consider the relation $\displaystyle R = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\}$ then $\displaystyle R$ is

On the sets of lines in a plane the Relation "is perpendicular to" is

On the set of lines, being perpendicular is a ______ relation.

Let $\displaystyle N$ be the set of all natural numbers; $\displaystyle E$ be the set of all even natural numbers then the function $\displaystyle f: N \to E$ defined $\displaystyle f(x) = 2x, x \in N$

If $\displaystyle A = \{1, 2, 3, 4\}$ and $\displaystyle B = \{1, 4, 9, 16, 25\}$ is a function $\displaystyle f$ is defined from set $\displaystyle A$ to $\displaystyle B$ where $\displaystyle f(x) = x^2$ then the range of $\displaystyle f$ is:

Identify the function from the following: (1,1), (1,2), (1,3)

If $\displaystyle A = \{a, b, c, d\}$, $\displaystyle B = \{p, q, r, s\}$ which of the following relation is a function from $\displaystyle A$ to $\displaystyle B$

If $\displaystyle f(n) = f(n-1) + f(n-2)$ when $\displaystyle n \ge 2$, $\displaystyle f(0) = 0$, $\displaystyle f(1) = 1$ then $\displaystyle f(7) = ?$

If $\displaystyle f(x) = \frac{x+1}{x-1}$, find $\displaystyle f^{-1}(x)$.

The inverse function $\displaystyle f^{-1}$ of $\displaystyle f(x) = 3y$ is:

Let $\displaystyle f: R \to R$ be defined by $\displaystyle f(x) = \begin{cases} 2x \text{ for } x \le 3 \\ x^2+1 \text{ for } 3 < x \le 6 \\ 3x \text{ for } x > 6 \end{cases}$ The value of $\displaystyle f(-1) + f(2) + f(4)$ is

If $\displaystyle f(x) = x^2-1$ and $\displaystyle g(x) = 2x+3$, then $\displaystyle [fog](3)-gof(-3)]$ is?

If $\displaystyle u(x) = \frac{1}{1-x}$, then $\displaystyle u^3(x)$ is:

The number of proper subsets of the set $\displaystyle \{3, 4, 5, 6, 7\}$ is

Let $\displaystyle A = \{1,2,3\}$ and $\displaystyle R = \{(1,1), (2,2), (3,3), (1,2)\}$ is

Let $\displaystyle A = \{1,2,3\}$ then the relation $\displaystyle R = \{(1,1), (2,3), (2,2), (3,3), (1,2)\}$ is called

On the set of lines, being perpendicular is a satisfies which property :

On the set of lines in a plane the Relation "is perpendicular to" is

$\displaystyle R = \{(1,1), (2,2), (3,3), (1,2), (2,3), (1,3)\}$ on the set $\displaystyle A = \{1,2,3\}$ is:

$\displaystyle A = \{1, 2, 3, 4, \dots, 10\}$ a relation on $\displaystyle A$, $\displaystyle R = \{(x, y) / x + y = 10, x \in A, y \in A\}$, then domain of $\displaystyle R^{-1}$ is

Consider the following relations on $\displaystyle A = \{1,2,3\}$. $\displaystyle R = \{(1,1), (1,2), (2,1), (2,2), (3,3)\}$, $\displaystyle S = \{(1,1), (1,2), (2,2), (2,3)\}$ and $\displaystyle \Phi = \text{empty set}$. Which one of these forms an equivalence relation?

If $\displaystyle f(x) = x^2$ and $\displaystyle g(x) = \sqrt{x}$, then

The range of the function $\displaystyle f$ defined by $\displaystyle f(x) = \sqrt{16-x^2}$ is

Let $\displaystyle A = R - \{3\}$ and $\displaystyle B = R - \{1\}$. Let $\displaystyle f: A \to B$ defined by $\displaystyle f(x) = \frac{x-2}{x-3}$. What is the value of $\displaystyle f^{-1}(\frac{1}{2})$?

If $\displaystyle f(x): N \to R$ is a function defined as $\displaystyle f(x) = 4x+3$, $\displaystyle \forall x \in N$, then $\displaystyle f^{-1}(x)$ is:

If $\displaystyle f(x) = x^2+x-1$ and $\displaystyle 4f(x) = f(2x)$, then find the value of 'x'.

If $\displaystyle f(x) = \frac{x}{1-x}$ & $\displaystyle g(x) = \frac{x}{1+x}$, then $\displaystyle gof(x)$ is

Let $\displaystyle f: R \to R$ be such that $\displaystyle f(x) = 2^x$, then $\displaystyle f(x+y)$ equals

Let $\displaystyle R$ be the set of real numbers such that the function $\displaystyle f: R \to R$ and $\displaystyle g: R \to R$ are defined by $\displaystyle f(x) = x^2+3x+1$ and $\displaystyle g(x) = 2x-3$. Find $\displaystyle (fog)$.

If $\displaystyle A = \{1,2,3,4\}$, $\displaystyle B = \{2,4,6,8\}$, $\displaystyle f: A \to B$ then $\displaystyle f^{-1}$ is:

If $\displaystyle f(x) = x^2-1$ and $\displaystyle g(x) = 2x+3$ then $\displaystyle gof(3)$

Find $\displaystyle g \circ f$ for the functions $\displaystyle f(x) = \sqrt{x}$, $\displaystyle g(x) = 2x^2+1$.

Find $\displaystyle f \circ g$ for the functions $\displaystyle f(x) = x^2$, $\displaystyle g(x) = 2x^2+1$.

If $\displaystyle f(x) = x+3$ and $\displaystyle g(x) = x^2$, then $\displaystyle fog(x)$

If $\displaystyle f(x) = \frac{x^2-4}{x-2}$ then $\displaystyle f(2)$ is.

If $\displaystyle f(x) = 3x^2+2x$ & $\displaystyle f(0) = 0$ then find $\displaystyle f(2)$.

If $\displaystyle f(x)=x^2-1$ and $\displaystyle g(x)=\frac{x+1}{2}$, then $\displaystyle \frac{f(3)}{f(3)+g(3)}$ is

If $\displaystyle f(x) = x^2-x$ and $\displaystyle f(1) = -10$ then the value of $\displaystyle K$ is

A function $\displaystyle f(x)$ is an even function, if

If $\displaystyle f(x)=\frac{2+x}{2-x}$, then $\displaystyle f^{-1}(x)$

If $\displaystyle f: R \to R$ is a function, defined by $\displaystyle f(x)=2^x$; then $\displaystyle f(x+y)$ is

If $\displaystyle f(x)=x+2, g(x)=7^x$ than $\displaystyle g \text{ of } (x)=$

Let R be a relation on N defined by $\displaystyle x+2y=8$. The domain of R is:

The domain of the function $\displaystyle f(x)=\frac{x^2+3x+5}{x^2-5x+4}$ is:

If $\displaystyle f(p)=\frac{1}{1-p}$, then $\displaystyle f^{-1}$ is

Determine $\displaystyle f(x)$, given that $\displaystyle f'(x)=12x^2-4x$ and $\displaystyle f(-3)=17$

If $\displaystyle f(x)=x^2-5$, evaluate $\displaystyle f(3)$, $\displaystyle f(-4)$, $\displaystyle f(5)$, and $\displaystyle f(1)$.

If $\displaystyle f(x)=\frac{x}{\sqrt{1+x^2}}$ and $\displaystyle g(x)=\frac{x}{\sqrt{1-x^2}}$ Find $\displaystyle fog?$

The range of the relation $\displaystyle \{(1,0)(2,0)(3,0)(4,0)(0,0)\}$ is

If $\displaystyle f(x)=x+2, g(x)=7^x$, then $\displaystyle go f(x)=$

If $\displaystyle f(x)=2x+2$ and $\displaystyle g(x)=x^2$, then the value of $\displaystyle fog(4)$ is:

Let R is the set of real numbers, such that the function $\displaystyle f: R \to R$ and $\displaystyle g: R \to R$ are defined by $\displaystyle f(x)=x^2+3x+1$ and $\displaystyle g(x)=2x-3$. Find $\displaystyle (fog):$

Let R is the set of real numbers such that the function $\displaystyle f: R \to R$ and $\displaystyle g: R \to R$ are defined by $\displaystyle f(x)=x^2+3x+1$ and $\displaystyle g(x)=2x-3$ Find $\displaystyle (fog):$

Given the function $\displaystyle f(x)=(2x+3)$, then the value of $\displaystyle f(2x)-2f(x)+3$ will be:

Find $\displaystyle f \circ g$ for the functions $\displaystyle f(x)=x^8, g(x)=2x^2+1$

If $\displaystyle f(x)=x+2, g(x)=7^x$, then $\displaystyle g \circ f(x)=$

$\displaystyle X=\{x,y,z,w\}; Y=\{1,2,3,4\}; H=\{(x,1);(y,2);(y,3);(z,4);(x,4)\}$

If $\displaystyle f = \{(2,2); (3,3); (4,4); (5,5); (6,6)\}$ be a relation on set $\displaystyle A = \{2,3,4,5,6\}$, then $\displaystyle f$ is:

If $\displaystyle f(y) = \frac{y-1}{y}$, find $\displaystyle f^{-1}(x)$.

If $\displaystyle A = \{1,2,3,4\}$ and $\displaystyle B = \{1,4,9,16,25\}$ is a function $\displaystyle f$ is defined set $\displaystyle A$ to $\displaystyle B$ where $\displaystyle f(x) = x^2$ then the range of $\displaystyle f$ is:

Which of the following relations is transitive but not reflexive for the set $\displaystyle S = \{3, 4, 6\}$?

If $\displaystyle A = \{1, 2, 3, 4\}$, $\displaystyle B = \{2, 4, 6, 8\}$ and $\displaystyle C = \{3, 4, 5, 6\}$ the value of $\displaystyle A - (B \cup C)$ is -

If $\displaystyle f(x) = \frac{x^2-4}{x-2}$, then $\displaystyle f(2)$ is

If $\displaystyle A = \{1, 2, 3, 4\}$ and $\displaystyle B = \{5, 6, 7, 6\}$, then cardinal number of the set $\displaystyle A \cap B$ is

If $\displaystyle A = \{1, 2, 3, 4, 5\}$ $\displaystyle B = \{2, 4\}$ and $\displaystyle C = \{1, 3, 5\}$ then $\displaystyle (A - C) \times B$ is

If $\displaystyle A = \{1, 2, 3, 4, 5\}$, $\displaystyle B = \{2, 4\}$ and $\displaystyle C = \{1, 3, 5\}$ then $\displaystyle (A - C) \times B$ is:

In a town of 20,000 families it was found that 40% families buy newspaper A, 20% families buy newspaper B and 10% families buy newspaper C, 5% families buy A and B, 3% buy B and C and 4% buy A and C. If 2% families buy all the three newspaper, then the number of families which buy A only is

Find the $\displaystyle f \circ g$ for the functions $\displaystyle f(x) = x^2$, $\displaystyle g(x) = x+1$.

If $\displaystyle A = \{2, 3\}$, $\displaystyle B = \{4, 5\}$ and $\displaystyle C = \{5, 6\}$ then the $\displaystyle A \times (B \cap C)$ is

Let $\displaystyle U$ be the universal set, $\displaystyle A$ and $\displaystyle B$ are the subsets of $\displaystyle U$. If $\displaystyle n(U) = 650$, $\displaystyle n(A) = 310$ $\displaystyle n(A \cap B) = 95$ and $\displaystyle n(B) = 190$, then $\displaystyle n(A \cap B)^c$ is equal to ($\displaystyle A$ and $\displaystyle B$ are the complement of $\displaystyle A$ and $\displaystyle B$ respectively):

The range of the function $\displaystyle f(x) = 3x - 2$ is

If $\displaystyle f(x)=1+\frac{1}{x}$, then the value of $\displaystyle f\left(f\left(\frac{1}{x}\right)\right)$ is

The function $\displaystyle f(x) = \frac{x^2 - 25}{x - 5}$ is undefined at $\displaystyle x=5$. What value must be assigned to f(5) if f(x) is to be continuous at $\displaystyle x=5$?

Let R is the set of real numbers such that the function $\displaystyle f: R \to R$ and $\displaystyle g: R \to R$ are defined by $\displaystyle f(x) = x^2 + 3x + 1$ and $\displaystyle g(x) = 2x - 3$ then $\displaystyle f \circ g(x)$ is:

For a relation R, aRb represents that a is related to b. If for all a, b, c, aRb and bRc gives that aRc, then the relation is

Let $\displaystyle S = \{a, b, c, d, e\}$. The number of non-empty proper subsets of S is

If $\displaystyle f(y) = \frac{1}{1+y}$ and $\displaystyle g(y) = \frac{y+1}{y}$ then $\displaystyle fog(y)=$

The inverse of the function $\displaystyle f(x) = \frac{2+3x}{x+5}$, by taking $\displaystyle f(x)$ as $\displaystyle y$, is