L6 - Sets A set is an unordered collection of objects...

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Sets A set is an unordered collection of objects Examples: {apple,orange,pear} is a set. {1,2,3,4,5} is a set. The objects of a set are called the elements (or members ) of a set. The set is said to contain its elements.
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Notation = The set of all real numbers = The set of all integers = The set of all positive integers {1,2,3,. ..} = The set of natural numbers {0,1,2,3,. ..} = The set of all rational numbers ="belongs to", or "is a member of" "does not belong to", "is not a member of" is the same as ( ) x A x A 
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Set builder notation We will often use set builder notation to define a set. We can write the set O of all positive odd integers as or Another example: We can define the set of all positive rational numbers as { | is an odd positive integer and 10} O x x x  { | odd and O x x x { | / for some , } x x p q p q 
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Equality of Sets Two sets A, B are equal if and only if they have the same elements. In such a case we write A=B. Two sets are equal if and only if Important notes: 1) The order of elements does not matter, so the sets {1,5,3} and {1,3,5} are equal. 2) It also does not matter if an element is listed more than once. So the sets {1,3,5} and {1,1,1,3,3,5,5,5,5,5} are equal. () x x A x B  
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Venn Diagrams Sets can be represented using Venn diagrams . In such a diagram, the universal set U, which represents all possible elements under consideration, is drawn with a large rectangle. Inside this rectangle, sets are drawn with circles or other shapes. Occasionally, elements are also marked on Venn diagrams as points.
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Venn Diagram The set of vowels in the alphabet is V={a,e,i,o,u}, which is depicted in the Venn diagram below
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Terms A set that has no elements is called the empty set , or the null set . It is denoted by { }, or Ex: The set of all irrational numbers that are also prime is the empty set. A set with one element is called a singleton set . .
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Subsets The set A is said to be a subset of the set B if and only if every element of A is also an element of B. denotes that A is a subset of B. Example: Consider again V={a,e,i,o,u}. A={a}, and B={o,u} are subsets of V. C={a,e,i,o,u,v} is not a subset of V. D={a,b} is also not a subset of V.
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L6 - Sets A set is an unordered collection of objects...

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