• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

Mathematics for Computing

Extracts from this document...

Introduction

IN1004 Mathematics for Computing Lecturer: Dr. Peter W.H. Smith peters@soi.city.ac.uk 1. Set Theory 1.1 Introduction A set is one of the most fundamental cornerstones of mathematics. It is a well-defined collection of objects. These objects are called elements and are said to be members of the set. Well-defined implies that we are able to determine whether it is the set under scrutiny. Thus we avoid sets based on opinion, e.g. the set of all great football players. 1.2 Notation and Set membership Capital letters, A,B,C... are used to represent sets and lowercase letters are used to represent elements. For a set A, we write x ????if x is an element of A; y ????indicates that y is not a member of A. A set can be designated by listing its elements within set braces. For example, if A is the set consisting of the first five positive integers, then we write A = {1,2,3,4,5}. In this example, 2 ?A but 6 ?A. Another standard notation for this set is A = {x | x is an integer and 1 ? x ? 5}. The vertical line | within the set may be read as "such that", the symbols {x |...} are read "the set of all x such that..." ...read more.

Middle

N = the set of non-negative integers of natural numbers = {0,1,2,3,...} c) Z+ = the set of positive integers = {1,2,3,...} d) Q = the set of rational numbers = {a/b | a,b ? Z, b ? 0} e) Q+ = the set of positive rational numbers = {r | r ? Q, r>0) f) R = the set of real numbers g) R+ = the set of positive real numbers 1.5 Set Operations and the Laws of Set Theory For A,B ? U we define the following: a) A ? B (the union of A and B) = {x | x ? A ? x ? B} b) A ? B ( the intersection of A and B) = {x | x ? A ? x ? B} c) A ? B ( the symmetric difference of A and B) = {x | (x ? A ? x ? B) ? x ? A ? B} = {x | x ? A ? B ? x ? A ? B} Note that if , A,B ? U then A ? B, A ? B, A ? B ? U .Consequently, ?, ? and ? are closed binary operations on P(U) or that P(U) is closed under these binary operations. Examples With U = {1,2,3,4,5,6,7,8,9,10}, A = {1,2,3,4,5}, B= {3,4,5,6,7} and C = {7,8,9} we have: a) ...read more.

Conclusion

Subsets of A containing 5 elements including 1,2 h) Proper subsets of A containing 1,2 i) Subsets of A with an even number of elements j) Subsets of A with an odd number of elements k) Subsets of A with an odd number of elements, including the element 3 6. For U= {1,2,3,4,5,6,7,8,9,10}, let A={1,2,3,4,5}, B={1,2,4,8}, C={1,2,3,5,7} and D={2,4,6,8}. Determine each of the following: a) (A ? B) ? C b) A ? (B ? C) __ __ c) C ? D ______ d) C ? D e) (A ? B) -C f) A ? (B - C) g) (B - C) - D h) B - (C - D) i) (A ? B) - (C ? D) 7. Let U= {a,b,c,...x,y,z} with A={a,b,c} and C={a,b,d,e}. If | A ? B| =2 and (A ? B) ? B ? C, determine B. 8. Using Venn diagrams or Membership Tables, investigate the truth or falsity of the each of the following for sets A,B,C ? U . a) A ? (B ? C) = (A ? B) ? (A ? C) b) A ? (B ? C) = (A ? B) ? (A ? C) 9. A supermarket discovers that from a sample of 50 shoppers, 30 buy tea, 25 buy coffee and 10 buy both coffee and tea. How many shoppers buy either coffee or tea.? (Hint - use Venn Diagrams) 1 Named after the English Logician John Venn (1834-1923) ?? ?? ?? ?? ...read more.

The above preview is unformatted text

This student written piece of work is one of many that can be found in our GCSE Accounting & Finance section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related GCSE Accounting & Finance essays

  1. Unit 5 Introduction to Accounting

    So it is wise to shop around the banks for the best deals to ensure that the expense is kept to the minimum. Overdraft, loan and mortgage interest Any form of bank borrowing will also incur interest charges that must be paid, the best advice is to compare the different

  2. Management Accounting Report.

    But the problem is would people want to buy the products of the same quality as before for more money? The company has to set out questionnaires to find it out. 4.1 Budgetary control part B Introduction (Budgetary control part B)

  1. What does Finance involve.

    So before I could work out my break even point I will need to work out my contribution: CONTRIBUTION = AVARAGE SELLING PRICE - AVARAGE VARIABLE COSTS = �18.25-�5.58 = �12.67 my contribution �12.67 is the amount of money the customers contribute towards my foxed cost every time they buy a product or use my service.

  2. This report has been produced as evidence for Unit 9 - 'Financial Services' - ...

    and are repayable over a period of time, usually between 6 months and 10 years. Lenders charge interest, which can either be fixed or variable, on the amount borrowed. This interest charge is expressed as an APR (annual percentage rate).

  1. The Basel Accord (Basel II)

    Generally speaking, these rules mean that the greater risk to which the bank is exposed, the greater the amount of capital the bank needs to hold to safeguard its solvency and overall economic stability (Basel Committee on Bank Supervision, 2003).

  2. Responsible accounting is the ability to conduct business in a way that is not ...

    If the cost of each product is the same, then there is no reason to incur the expense and paper work necessary to keep track of how much each order costs to produce. Under process costing, instead of keeping track of the cost of each order separately, one calculates the

  1. My hypothesis is that the top 3 sets (A, B and C) predict both ...

    Truncating would mean I would take the whole number and cut of the decimal places. I realised that this will not be very useful as I will never get the highest number. E.g. If there are 12 numbers, the highest random number is 0.999 and if I multiply that by

  2. Project DR: Manufacturing Division

    Service sells and Manufacturing receives credit d) Calculate the Manufacturing division break-even sales level e) Separate Fixed and Variable costs, and define "why" Allocation, Break-Even, and Costs: Exhibit 1 focuses on meeting the revenue goals. The breakeven point analysis is essential because it shows at what point the operating income is zero.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work