Function
Q1.
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 \varepsilonWhich of the following statements is/are TRUE?Q2.
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.Q3.
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]Q4.
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 ______.Q5.
Let A be a finite set having x elements and let B be a finite set having y elements. What is the number of distinct functions mapping B into A.Q6.
Let f: B \rightarrow C and g: A \rightarrow B be two functions and let h = f\cdotg. Given that h is an onto function which one of the following is TRUE?Q7.
Suppose X and Y are sets and |X| and |Y| are their respective cardinality. It is given that there are exactly 97 functions from X to Y. From this one can conclude thatQ8.
Consider the function y=|x| in the interval [-1, 1]. In this interval, the function isQ9.
Let A and B be sets with cardinalities m and n respectively. The number of one-one mappings from A to B, when m \lt n, isQ10.
How many onto (or surjective) functions are there from an n-element ( n \geq 2 ) set to a 2-element set?