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_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?
F is NOT well-defined. | |
F is an onto (or surjective) function. | |
F is a one-to-one (or injective) function. | |
F is a bijective function. |
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_n= \left \{ \begin{matrix} n+1, &n \text{ is odd} \\ 1,& \text{otherwise} \end{matrix} \right.
\frac{x(1+x^2)}{(1-x^2)^2}+\frac{1}{1-x} | |
\frac{x(3-x^2)}{(1-x^2)^2}+\frac{1}{1-x} | |
\frac{2x}{(1-x^2)^2}+\frac{1}{1-x} | |
\frac{x}{(1-x^2)^2}+\frac{1}{1-x} |
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]
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]
There does not exist a bijection from S1 to S2 | |
There exists a surjection from S1 to S2 | |
There exists a bijection from S1 to S2 | |
There does not exist an injection from S1 to S2 |
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 ______.
16 | |
17 | |
15 | |
14 |
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:
\frac{h(x)}{g(x)} | |
\frac{-1}{x} | |
\frac{g(x)}{h(x)} | |
\frac{x}{(1-x)^{2}} |
Question 5 Explanation:
There are 5 questions to complete.
Q1 is from calculus, kindly move it from here to the calculus section
We have moved Q1 to calculus.