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?

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.