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Mathematics LibreTexts

Test 1

( \newcommand{\kernel}{\mathrm{null}\,}\)

These mock exams are provided to help you prepare for Term/Final tests. The best way to use these practice tests is to try the problems as if you were taking the test. Please don't look at the solution until you have attempted the question(s). Only reading through the answers or studying them will typically not be helpful in preparing since it is too easy to convince yourself that you understand them.

Exercise 1

Define a relation  on N by:

ab if and only if a2+b is even.

  1.  [4 marks] Prove that  defines an equivalence relation.
  2.  [2 marks] What is the equivalence class of 3?
Answer

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Exercise 2
  1. [6 marks] Show that 17 and 60 are relatively prime, determine multuplicative inverse of [17] in Z60.
  2. [4 marks] Find  all xZ satisfying the equation 17x12(mod60).
Answer

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Exercise 3

[5 marks] If fg is one-to-one then g is one-to-one.

[5 marks] If fg is one-to-one and g is onto, then f is one-to-one.

Answer

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Exercise 4
  1. [5 marks] Prove or disprove: U(10) is cyclic group.
  2. [5 marks] Show that H={0,2,4,6} is a subgroup of Z8, using the operation +mod(8).
Answer

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Exercise 5

[10 marks] Let G be a group. Prove or disprove the following statements.

  1.  If G is abelian, then G is cyclic.
  2.  If G is cyclic, then G is abelian.

 


Test 1 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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