Sets

Basic properties of sets



Two characteristics of a set

  1. There are no repeated occurrences of elements
  2. There is no particular order or arrangement of the elements


How do you show A is a subset of B?
If A and B are sets and every element of A is also an element of B, then A is a subset of B.

How do you show A = B?
A is a subset of B, and B is a subset of A

Operations on sets

  • Union - A union B = set of x s.t. x is an element of A or x is an element of B
  • Intersection - A intersect B = set of x s.t. x is an element of A and x is an element of B 
  • Complement

Cardinality

Inclusion-Exclusion principle
|A ∪ B| = |A| + |B| − |A ∩ B|.

Inclusion-Exclusion for 3 sets
|A ∪ B ∪ C| = |A| + |B| + |C| − |B ∩ C| − |A ∩ B| − |A ∩ C| + |A ∩ B ∩ C|

Difference rule
|A − B| = |A| − |A ∩ B|.




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