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Mathematics for Computing

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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.

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