# Functions

 Question 1
Let $f:A\rightarrow B$ be an onto (or surjective) function, where $A$ and $B$ are nonempty sets. Define an equivalence relation $\sim$ on the set $A$ as
$a_1\sim a_2 \text{ if } f(a_1)=f(a_2),$
where $a_1, a_2 \in A$ . Let $\varepsilon =\{[x]:x \in A\}$ be the set of all the equivalence classes under $\sim$. Define a new mapping $F: \varepsilon \rightarrow B$ as
$F([x]) = f(x),$ for all the equivalence classes $[x]$ in $\varepsilon$
Which of the following statements is/are TRUE?
 A F is NOT well-defined. B F is an onto (or surjective) function. C F is a one-to-one (or injective) function. D F is a bijective function.
GATE CSE 2023   Discrete Mathematics
Question 1 Explanation:
 Question 2
Which one of the following is the closed form for the generating function of the sequence $\{a_n \}_{n \geq 0}$ defined below?
$a_n= \left \{ \begin{matrix} n+1, &n \text{ is odd} \\ 1,& \text{otherwise} \end{matrix} \right.$
 A $\frac{x(1+x^2)}{(1-x^2)^2}+\frac{1}{1-x}$ B $\frac{x(3-x^2)}{(1-x^2)^2}+\frac{1}{1-x}$ C $\frac{2x}{(1-x^2)^2}+\frac{1}{1-x}$ D $\frac{x}{(1-x^2)^2}+\frac{1}{1-x}$
GATE CSE 2022   Discrete Mathematics
Question 2 Explanation:

 Question 3
Consider the following sets, where $n\geq 2$:

S1: Set of all nxn matrices with entries from the set $\{a, b, c\}$
S2: Set of all functions from the set$\{0,1,2,...,n^2-1\}$ to the set $\{0, 1, 2 \}$

Which of the following choice(s) is/are correct?
[MSQ]
 A There does not exist a bijection from S1 to S2 B There exists a surjection from S1 to S2 C There exists a bijection from S1 to S2 D There does not exist an injection from S1 to S2
GATE CSE 2021 SET-2   Discrete Mathematics
Question 3 Explanation:
 Question 4
If the ordinary generating function of a sequence $\{a_{n}\}_{n=0}^{\infty} \; is \; \frac{1+z}{(1-z)^{3}}$ then $a_{3}-a_{0}$ is equal to ______.
 A 16 B 17 C 15 D 14
GATE CSE 2017 SET-2   Discrete Mathematics
Question 4 Explanation:
 Question 5
If $g(x)=1-x$ and $h(x)=\frac{x}{x-1}$, then $\frac{g(h(x))}{h(g(x))}$ is:
 A $\frac{h(x)}{g(x)}$ B $\frac{-1}{x}$ C $\frac{g(x)}{h(x)}$ D $\frac{x}{(1-x)^{2}}$
GATE CSE 2015 SET-1   Discrete Mathematics
Question 5 Explanation:

There are 5 questions to complete.

### 2 thoughts on “Functions”

1. Q1 is from calculus, kindly move it from here to the calculus section

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