• 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...


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.


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.


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

    There is often a different charge a separate charge for each of these items and since large organisations carry out many transactions each day and may pay in many cheques and/ or lots of cash and coin most days, these charges can run into hundreds of pounds every month.

  2. Project DR: Manufacturing Division

    Allocating costs is a time consuming process. While this list is not all-inclusive, it serves to show what miscommunication, lack of knowledge, and unsupportive management can do to a system that works -- if handled improperly. This is why it is so important that the management of Diversified Resources stands behind cost allocations, and explain the true meaning and purposes of allocation.

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

    I will be looking a various rates of APR (interest rates) Jamie would be paying when he takes out a mortgage. There are a range of choices from which Jamie can look at, they include: 1. Fixed rates: The amount you pay per month is fixed over period of time over one year or more.

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

    Process costing is used when identical goods or services are mass- produced or produced in a continuous flow. For example, the television sets that Sony sells to K-Mart are the same as the television sets sold to Wal-Mart or Target.

  1. The Basel Accord (Basel II)

    In practice, Basel II attempts to accomplish this by setting up rigorous risk and capital management requirements designed to ensure that a bank holds capital reserves appropriate to the risk the bank exposes itself to through its lending and investment practices.

  2. 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

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