Math135Lecture06StudentNotes-1

Math135Lecture06StudentNotes-1 - Math 135 Lecture 6 Sets...

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Unformatted text preview: Math 135: Lecture 6: Sets Definition 6.1. A set is a collection of objects. The objects that make up a set are called its elements (or members ). Sets can contain any type of objects. • Use uppercase letters ( A,B,C ... ) to represent sets. • Use lowercase letters ( a,b,c,... ) to represent members. • If a is an element of the set A , we write • If a is not an element of the set A , we write Example 6.2. The set of primes less than 10 is Many sets are either too large to be listed or are defined by a rule. In these cases, we employ set-builder notation which makes use of a defining property of the set. Example 6.3. The set of all real numbers between 1 and 2 could be written as { x ∈ R | 1 ≤ x ≤ 2 } Common Sets N Z Q Q R C Subsets A set A is called a subset of a subset B , and written A ⊆ B , if every element of A belongs to B . Symbolically, we write A ⊆ B means x ∈ A ⇒ x ∈ B or equivalently A ⊆ B means For all x ∈ A, it is also true that x ∈ B A set A is called a proper subset of a subset B , and written A ⊂ B , if every element of A belongs to B and there exists an element in B which does not belong to A . Example 6.4. { 1 , 2 , 3 } { 1 , 2 , 3 , 4 } Supersets A set A is called a superset of a subset B , and written A ⊇ B , if every element of B belongs to A . A ⊇ B is equivalent to B ⊆ A . A set A is called a proper superset of a subset B , and written A ⊃ B , if every element of B belongs to A and there exists an element in...
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Math135Lecture06StudentNotes-1 - Math 135 Lecture 6 Sets...

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