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how to prove function injective or not
Let A, B, and C be finite sets, and f : B to C and g : A to B be functions. Let h be the function with domain A and range C that maps x in A to f (g(x)). Prove or disprove the following claim: If h is injective, then g must be injective.
asked
Dec 21, 2020
in
Set Theory & Algebra
by
tyagiabhi
(
5
points)

20
views
functions
discretemaths
0
votes
0
answers
Gatebook function
How to solve this type of question ?
asked
Oct 2, 2020
in
Set Theory & Algebra
by
Raj_81
(
23
points)

23
views
discretemaths
functions
0
votes
0
answers
Discrete MathematicsFunctions
Let $A=\{1,2,3,4,5\},B=\{w,x,y,z\},A_{1}=\{2,3,5\}\subseteq A$ and $g:A_{1}\rightarrow B.$ In how many ways can $g$ be extended to a function $f:A\rightarrow B$
asked
Jul 25, 2020
in
Set Theory & Algebra
by
KUSHAGRA गुप्ता
(
1.4k
points)

45
views
discretemathematics
functions
0
votes
0
answers
GATE2015139 Video Solution
Consider the operations $\textit{f (X, Y, Z) = X'YZ + XY' + Y'Z'}$ and $\textit{g (X, Y, Z) = X'YZ + X'YZ' + XY}$ Which one of the following is correct? Both $\left\{\textit{f} \right\}$ and $\left\{ \textit{g}\right\}$ ... $\left\{ \textit{g}\right\}$ is functionally complete Neither $\left\{ \textit{f}\right\}$ nor $\left\{\textit{g}\right\}$ is functionally complete
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

17
views
gate20151
settheory&algebra
functions
difficult
videosolution
0
votes
0
answers
GATE2015240 Video Solution
The number of onto functions (surjective functions) from set $X = \{1, 2, 3, 4\}$ to set $Y=\{a,b,c\}$ is ______.
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

5
views
gate20152
settheory&algebra
functions
normal
numericalanswers
videosolution
0
votes
0
answers
GATE2016128 Video Solution
A function $f: \Bbb{N^+} \rightarrow \Bbb{N^+}$ , defined on the set of positive integers $\Bbb{N^+}$,satisfies the following properties: $f(n)=f(n/2)$ if $n$ is even $f(n)=f(n+5)$ if $n$ is odd Let $R=\{ i \mid \exists{j} : f(j)=i \}$ be the set of distinct values that $f$ takes. The maximum possible size of $R$ is ___________.
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

17
views
gate20161
settheory&algebra
functions
normal
numericalanswers
videosolution
0
votes
0
answers
GATE2015226 Video Solution
Let $f(x)=x^{\left(\frac{1}{3}\right)}$ and $A$ denote the area of region bounded by $f(x)$ and the Xaxis, when $x$ varies from $1$ to $1$. Which of the following statements is/are TRUE? $f$ is continuous in $[1, 1]$ $f$ is not bounded in $[1, 1]$ $A$ is nonzero and finite II only III only II and III only I, II and III
asked
Apr 18, 2020
in
Calculus
by
admin
(
573
points)

8
views
gate20152
continuity
functions
normal
videosolution
0
votes
0
answers
GATE2014349 Video Solution
Consider the set of all functions $f:\{0,1, \dots,2014\} \to \{0,1,\dots, 2014\}$ such that $ f\left(f\left(i\right)\right)=i$, for all $0 \leq i \leq 2014$. Consider the following statements: $P$. For each such function it must be the case that for every ... is CORRECT? $P, Q$ and $R$ are true Only $Q$ and $R$ are true Only $P$ and $Q$ are true Only $R$ is true
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

7
views
gate20143
settheory&algebra
functions
normal
videosolution
0
votes
0
answers
GATE201432 Video Solution
Let $X$ and $Y$ be finite sets and $f:X \to Y$ be a function. Which one of the following statements is TRUE? For any subsets $A$ and $B$ of $X, fA \cup B = f(A) + f(B)$ For any subsets $A$ and $B$ of $X, f(A \cap B) = f(A) \cap f(B)$ For any subsets $A$ ... $S$ and $T$ of $Y, f^{1}(S \cap T) = f^{1}(S) \cap f^{1}(T)$
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

8
views
gate20143
settheory&algebra
functions
normal
videosolution
0
votes
0
answers
GATE2014150 Video Solution
Let ܵ$S$ denote the set of all functions $f:\{0,1\}^4 \to \{0,1\}$. Denote by $N$ the number of functions from S to the set $\{0,1\}$. The value of $ \log_2 \log_2N $ is _______.
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

7
views
gate20141
settheory&algebra
functions
combinatory
numericalanswers
videosolution
0
votes
0
answers
GATE20012.3 Video Solution
Let $f: A \rightarrow B$ a function, and let E and F be subsets of $A$. Consider the following statements about images. $S1: f(E \cup F) = f(E) \cup f(F)$ $S2: f(E \cap F)=f(E) \cap f(F)$ Which of the following is true about S1 and S2? Only S1 is correct Only S2 is correct Both S1 and S2 are correct None of S1 and S2 is correct
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

13
views
gate2001
settheory&algebra
functions
normal
videosolution
0
votes
0
answers
GATE200625 Video Solution
Let $S = \{1, 2, 3,\ldots, m\}, m >3.$ Let $X_1,\ldots,X_n$ be subsets of $S$ each of size $3.$ Define a function $f$ from $S$ to the set of natural numbers as, $f(i)$ is the number of sets $X_j$ that contain the element $i.$ That is $f(i)=\left  \left\{j \mid i\in X_j \right\} \right$ then $ \sum_{i=1}^{m} f(i)$ is: $3m$ $3n$ $2m+1$ $2n+1$
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

7
views
gate2006
settheory&algebra
normal
functions
videosolution
0
votes
0
answers
GATE200337 Video Solution
Let \(f : A \to B\) be an injective (onetoone) function. Define \(g : 2^A \to 2^B\) as: \(g(C) = \left \{f(x) \mid x \in C\right\} \), for all subsets $C$ of $A$. Define \(h : 2^B \to 2^A\) as: \(h(D) = \{x \mid x \in A, f(x) \in D\}\), for all ... always true? \(g(h(D)) \subseteq D\) \(g(h(D)) \supseteq D\) \(g(h(D)) \cap D = \phi\) \(g(h(D)) \cap (B  D) \ne \phi\)
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

14
views
gate2003
settheory&algebra
functions
normal
videosolution
0
votes
0
answers
GATE200543 Video Solution
Let $f: B \to C$ and $g: A \to B$ be two functions and let $h = f o g$. Given that $h$ is an onto function which one of the following is TRUE? $f$ and $g$ should both be onto functions $f$ should be onto but $g$ need not to be onto $g$ should be onto but $f$ need not be onto both $f$ and $g$ need not be onto
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

6
views
gate2005
settheory&algebra
functions
normal
videosolution
0
votes
0
answers
GATE2015254 Video Solution
Let $X$ and $Y$ denote the sets containing 2 and 20 distinct objects respectively and $F$ denote the set of all possible functions defined from $X$ to $Y$. Let $f$ be randomly chosen from $F$. The probability of $f$ being onetoone is ______.
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

9
views
gate20152
settheory&algebra
functions
normal
numericalanswers
videosolution
0
votes
0
answers
GATE201237 Video Solution
How many onto (or surjective) functions are there from an $n$element $(n ≥ 2)$ set to a $2$element set? $ 2^{n}$ $2^{n} – 1$ $2^{n} – 2$ $2(2^{n} – 2)$
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

10
views
gate2012
settheory&algebra
functions
normal
videosolution
0
votes
0
answers
GATE19962.1 Video Solution
Let $R$ denote the set of real numbers. Let $f:R\times R \rightarrow R \times R$ be a bijective function defined by $f(x,y) = (x+y, xy)$. The inverse function of $f$ is given by $f^{1} (x,y) = \left( \frac {1}{x+y}, \frac{1}{xy}\right)$ ... $f^{1}(x,y)=\left [ 2\left(xy\right),2\left(x+y\right) \right ]$
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

7
views
gate1996
settheory&algebra
functions
normal
videosolution
0
votes
0
answers
GATE2005IT31 Video Solution
Let $f$ be a function from a set $A$ to a set $B$, $g$ a function from $B$ to $C$, and $h$ a function from $A$ to $C$, such that $h(a) = g(f(a))$ for all $a ∈ A.$ Which of the following statements is always true for all such functions $f$ ... is onto $h$ is onto $\implies$ $f$ is onto $h$ is onto $\implies$ $g$ is onto $h$ is onto $\implies$ $f$ and $g$ are onto
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

10
views
gate2005it
settheory&algebra
functions
normal
videosolution
0
votes
0
answers
GATE200339 Video Solution
Let $\Sigma = \left\{a, b, c, d, e\right\}$ be an alphabet. We define an encoding scheme as follows: $g(a) = 3, g(b) = 5, g(c) = 7, g(d) = 9, g(e) = 11$. Let $p_i$ denote the ith prime number $\left(p_1 = 2\right)$ ... numbers is the encoding, $h$, of a nonempty sequence of strings? $2^73^75^7$ $2^83^85^8$ $2^93^95^9$ $2^{10}3^{10}5^{10}$
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

8
views
gate2003
settheory&algebra
functions
normal
videosolution
0
votes
0
answers
GATE20153GA5 Video Solution
A function $f(x)$ is linear and has a value of 29 at $x=2$ and 39 at $x=3$. Find its value at $x=5$. $59$ $45$ $43$ $35$
asked
Apr 18, 2020
in
Numerical Ability
by
admin
(
573
points)

8
views
gate20153
numericalability
normal
functions
videosolution
0
votes
0
answers
GATE20073 Video Solution
What is the maximum number of different Boolean functions involving $n$ Boolean variables? $n^2$ $2^n$ $2^{2^n}$ $2^{n^2}$
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

10
views
gate2007
combinatory
functions
normal
videosolution
0
votes
0
answers
GATE19961.3 Video Solution
Suppose $X$ and $Y$ are sets and $X \text{ and } Y$ are their respective cardinality. It is given that there are exactly $97$ functions from $X$ to $Y$. From this one can conclude that $X =1, Y =97$ $X =97, Y =1$ $X =97, Y =97$ None of the above
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

8
views
gate1996
settheory&algebra
functions
normal
videosolution
0
votes
0
answers
GATE20152GA9 Video Solution
If p, q, r, s are distinct integers such that: $f (p, q, r, s) = \text{ max } (p, q, r, s)$ $g (p, q, r, s) = \text{ min } (p, q, r, s)$ ... operations are valid with two variable functions of the form $f(p, q)$ What is the value of $fg \left(h \left(2, 5, 7, 3\right), 4, 6, 8\right)$?
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

7
views
gate20152
settheory&algebra
functions
normal
numericalanswers
videosolution
0
votes
0
answers
GATE20152GA3 Video Solution
Consider a function $f(x) = 1 x \text{ on } 1 \leq x \leq 1$. The value of $x$ at which the function attains a maximum, and the maximum value of the function are: $0, 1$ $1, 0$ $0, 1$ $1, 2$
asked
Apr 18, 2020
in
Calculus
by
admin
(
573
points)

9
views
gate20152
settheory&algebra
functions
normal
maximaminima
videosolution
0
votes
0
answers
GATE199713 Video Solution
Let $F$ be the set of onetoone functions from the set $\{1, 2, \dots, n\}$ to the set $\{1, 2,\dots, m\}$ where $m\geq n\geq1$. How many functions are members of $F$? How many functions $f$ in $F$ satisfy the property $f(i)=1$ for some $i, 1\leq i \leq n$? How many functions $f$ in $F$ satisfy the property $f(i)<f(j)$ for all $i,j \ \ 1\leq i \leq j \leq n$?
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

12
views
gate1997
settheory&algebra
functions
normal
descriptive
videosolution
0
votes
0
answers
GATE19968 Video Solution
Let $F$ be the collection of all functions $f: \{1, 2, 3\} \to \{1, 2, 3\}$. If $f$ and $g \in F$, define an equivalence relation $\sim$ by $f\sim g$ if and only if $f(3) = g(3)$. Find the number of equivalence classes defined by $\sim$. Find the number of elements in each equivalence class.
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

6
views
gate1996
settheory&algebra
relations
functions
normal
descriptive
videosolution
0
votes
0
answers
GATE20062 Video Solution
Let $X,Y,Z$ be sets of sizes $x, y$ and $z$ respectively. Let $W = X \times Y$ and $E$ be the set of all subsets of $W$. The number of functions from $Z$ to $E$ is $z^{2^{xy}}$ $z \times 2^{xy}$ $z^{2^{x+y}}$ $2^{xyz}$
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

5
views
gate2006
settheory&algebra
normal
functions
videosolution
0
votes
0
answers
GATE2006IT6 Video Solution
Given a boolean function $f (x_1, x_2, \ldots, x_n),$ which of the following equations is NOT true? $f (x_1, x_2, \ldots, x_n) = x_1'f(x_1, x_2, \ldots, x_n) + x_1f(x_1, x_2, \ldots, x_n)$ ... $f (x_1, x_2, \ldots , x_n) = f(0, x_2, , x_n) + f(1, x_2, \ldots, x_n)$
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

5
views
gate2006it
settheory&algebra
functions
normal
videosolution
0
votes
0
answers
GATE20153GA8 Video Solution
Choose the most appropriate equation for the function drawn as thick line, in the plot below. $x=yy$ $x=(yy)$ $x=y+y$ $x=(y+y)$
asked
Apr 18, 2020
in
Numerical Ability
by
admin
(
573
points)

6
views
gate20153
numericalability
normal
functions
videosolution
0
votes
0
answers
GATE19981.8 Video Solution
The number of functions from an $m$ element set to an $n$ element set is $m + n$ $m^n$ $n^m$ $m*n$
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

4
views
gate1998
settheory&algebra
combinatory
functions
easy
videosolution
0
votes
0
answers
GATE201515 Video Solution
If $g(x) = 1  x$ and $h(x) = \frac{x}{x1}$, then $\frac{g(h(x))}{h(g(x))}$ is: $\frac{h(x)}{g(x)}$ $\frac{1}{x}$ $\frac{g(x)}{h(x)}$ $\frac{x}{(1x)^{2}}$
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

9
views
gate20151
settheory&algebra
functions
normal
videosolution
0
votes
0
answers
GATE19938.6 Video Solution
Let $A$ and $B$ be sets with cardinalities $m$ and $n$ respectively. The number of oneone mappings from $A$ to $B$, when $m < n$, is $m^n$ $^nP_m$ $^mC_n$ $^nC_m$ $^mP_n$
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

8
views
gate1993
settheory&algebra
functions
easy
videosolution
0
votes
0
answers
GATE19871II Video Solution
The total number of Boolean functions which can be realised with four variables is: $4$ $17$ $256$ $65, 536$
asked
Apr 18, 2020
in
Digital Logic
by
admin
(
573
points)

8
views
gate1987
digitallogic
booleanalgebra
functions
combinatory
videosolution
0
votes
0
answers
GATE19879b Video Solution
How many onetoone functions are there from a set $A$ with $n$ elements onto itself?
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

9
views
gate1987
settheory&algebra
functions
descriptive
videosolution
0
votes
0
answers
GATE2018 CH: GA3 Video Solution
For $0\leq{x}\leq{2\pi}$, $\sin x \text{ and } \cos x$ are both decreasing functions in the interval _________ . $\left(0,\dfrac{\pi}{2}\right)$ $\left(\dfrac{\pi}{2},\pi\right)$ $\left(\pi,\dfrac{3\pi}{2}\right)$ $\left(\dfrac{3\pi}{2},2\pi\right)$
asked
Apr 18, 2020
in
Numerical Ability
by
admin
(
573
points)

15
views
gate2018ch
numericalability
functions
trigonometry
videosolution
0
votes
0
answers
GATE2016 ME2: GA10 Video Solution
Which of the following curves represents the function $y=\ln \left( \mid e^{\left[\mid \sin \left( \mid x \mid \right) \mid \right]} \right)$ for $\mid x \mid < 2\pi$? Here, $x$ represents the abscissa and $y$ represents the ordinate.
asked
Apr 18, 2020
in
Numerical Ability
by
admin
(
573
points)

12
views
gate2016me2
functions
numericalability
videosolution
0
votes
0
answers
GATE198913c Video Solution
Find the number of single valued functions from set A to another set B, given that the cardinalities of the sets A and B are $m$ and $n$ respectively.
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

7
views
gate1989
descriptive
functions
videosolution
0
votes
0
answers
GATE198813ii Video Solution
If the set $S$ has a finite number of elements, prove that if $f$ maps $S$ onto $S$, then $f$ is onetoone.
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

7
views
gate1988
descriptive
settheory&algebra
functions
videosolution
0
votes
0
answers
GATE20014 Video Solution
Consider the function $h: N \times N \rightarrow N$ so that $h(a,b) = (2a +1)2^b  1$, where $N=\{0,1,2,3,\dots\}$ is the set of natural numbers. Prove that the function $h$ is an injection (oneone). Prove that it is also a Surjection (onto)
asked
Apr 18, 2020
in
Set Theory & Algebra
by
admin
(
573
points)

6
views
gate2001
functions
settheory&algebra
normal
descriptive
videosolution
0
votes
0
answers
GATE2012 AR: GA7 Video Solution
Let $f(x) = x – [x],$ where $x\geq 0$ and $[x]$ is the greatest integer not larger than $x.$ Then $f(x)$ is a monotonically increasing function monotonically decreasing function linearly increasing function between two integers linearly decreasing function between two integers
asked
Apr 18, 2020
in
Numerical Ability
by
admin
(
573
points)

6
views
gate2012ar
numericalability
functions
normal
videosolution
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