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Unformatted text preview: Chapter 5: Set Theory February 4, 2008 Outline 1 5.1 Basic Definitions of Set Theory 2 5.2 Properties of Sets 3 5.3 Disproofs and Algebraic Proofs Basic Definitions • As we have mentioned earlier, a set is a collection of objects. These objects are called elements of the set. To indicate that an object x is an element of a set S , we write x ∈ S and the fact that x is not an element of S , we write it as x 6∈ S • The ordering of elements in a set is irrelevant; i.e. { a , b , c } = { b , c , a } • Also, an element of a set can be listed more than once { a , a , b , c , c , c } = { a , b , c } • We have to be careful when working with sets and their elements. For example, { a } 6 = a The reason for this is that { a } is a set whose element is a , while a is simply an object. • A set can be an element of another set; for example, { a , b , { a , c }} is the set whose elements are a , b , and { a , c } . • We have also mentioned that a set can be defined by some predicate; e.g. T = { x ∈ S  P ( x ) } means that the set T consists of all those elements of some set S which satisfy the property (predicate) P ( x ) . Examples (a) T = { x ∈ Z  x is divisible by 2 } is a set of all even numbers. (b) T = { x ∈ R  < x ≤ 2 } is the interval ( , 2 ] of the real line. (c) T = { x ∈ Z  < x ≤ 2 } is the set { 1 , 2 } . Subsets Definition If A and B are sets, then we say that A is a subset of B , which we write as A ⊆ B if and only if, every element of the set A is also an element of the set B . A ⊆ B ⇔ ∀ x , x ∈ A → x ∈ B We can also say that A is contained in B . • So, A 6⊆ B ⇔ ∃ x such that x ∈ A ∧ x 6∈ B Example Given the sets A = { a , b , c , d } , B = { a , c } , C = { d , c } we see that 1 B ⊆ A 2 C ⊆ A 3 B 6⊆ C and C 6⊆ B • Clearly, for any set S , S ⊆ S Definition Suppose A and B are two sets. We say that A is a proper subset of B , and we write it as A ⊂ B if A is a subset of B , but there is at least one element of B which is not in A . Examples (a) { a , b } ⊂ { a , b , c } (b) Z ⊂ Q ⊂ R ⊂ C • As we have seen, sets may have other sets as their elements, but this is not the same thing as being a subset. Example Consider the set S = { , 1 , { 2 , 3 }} Then, { 2 , 3 } ∈ S , { 2 , 3 } 6⊆ S On the other hand, { , 1 } ⊆ S , { , 1 } 6∈ S Definition We say that two sets A and B are equal if they have the same elements A = B ⇔ ∀ x , x ∈ A if and only if x ∈ B • Clearly, A = B ⇔ A ⊆ B and B ⊆ A Example Let A , B , C , and D be defined as follows: A = { n ∈ Z  n = 2 p for some integer p } B = the set of all even integers C = { m ∈ Z  m = 2 q 2 for some integer q } D = { k ∈ Z  k = 3 r + 1 for some integer r } Is (a) A = B ?...
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This note was uploaded on 04/17/2008 for the course MTH 314 taught by Professor Forgot during the Spring '08 term at Ryerson.
 Spring '08
 forgot
 Algebra, Set Theory, Sets

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