Unit 1-Lecture 6：Sets and Sequence

1 Set Definition

• A set is a bunch of objects, which are called the elements of the set.
• Set builder notation is to define a set using predicate: the set consists of all values that make the predicate true.

2 Set Operation

• The union of set A and B:
  
  x∈A∪BIFFx∈AORx∈B
• The intersection of set A and B:
  
  x∈A∩BIFFx∈AANDx∈B
• The set difference of A and B:
  
  x∈A−BIFFx∈AANDx∉B
• The complement of set A:
  
  x∈A¯IFFx∈DANDx∉A
• The power of set A:
  
  B∈POW(A)IFFB⊂A

3 Set Equalities

• X = Y means that z∈x if and only if y z∈z .
• Distributive Law for Sets:
  
  A∩(B∪C)=(A∩B)∪(A∩C)
• De Morgan’s Law for set equality:
  
  A∩B¯¯¯¯¯¯¯¯¯=A¯¯¯∪B¯¯¯

4 Sequence

• Definition: a list of objects called terms or components.
• The difference between Set and Sequence:
  
  • The elements of a set are required to be distinct, but terms in a sequence can be the same.
  • The terms in a sequence have a specified order, but the elements of a set do not.

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Reference

[1] Lehman E, Leighton F H, Meyer A R. Mathematics for Computer Science[J]. 2015.