Set Theorem By Derek Mok, Alex Yau, Henry Tsang and Tommy Lee

Definition and Info (History) Any collection of distinct things considered as a whole Any collection of distinct things considered as a whole Invented at the end of the 19 th centaury Invented at the end of the 19 th centaury Set theory can be viewed as the foundation upon which nearly all of mathematics can be built and the source from which nearly all mathematics can be derived Set theory can be viewed as the foundation upon which nearly all of mathematics can be built and the source from which nearly all mathematics can be derived

Element of Sets Object of a set = “elements” or “members” Object of a set = “elements” or “members” The element of a set can be anything: The element of a set can be anything: Numbers Numbers People People Letters Letters Alphabet Alphabet Other sets Other sets

Describing Sets Sets can be defined in several ways Sets can be defined in several ways Defined using words defined by explicitly listing its elements between braces When two description define the same set, such as A is identical to C, we can write A=C to express this equality When two description define the same set, such as A is identical to C, we can write A=C to express this equality Ellipses (…) indicates that the list continues in an obvious way Ellipses (…) indicates that the list continues in an obvious way

Set membership When something IS an element of a particular set, then this is symbolized by When something IS an element of a particular set, then this is symbolized by When something IS NOT an element of a particular set, then this is symbolized by When something IS NOT an element of a particular set, then this is symbolized by Example: Example: and since 285 = 17² − 4; but and since 285 = 17² − 4; but and and

Cardinality of a Set Each of the sets given above has a definite number of members. Each of the sets given above has a definite number of members. However a set can also have zero members, called an empty set, represented by ø However a set can also have zero members, called an empty set, represented by ø Example: The set A of all three-sided square has zero members, therefore A = ø Example: The set A of all three-sided square has zero members, therefore A = ø A set can also have an infinite numbers of members A set can also have an infinite numbers of members Example: the set of natural numbers Example: the set of natural numbers

Subsets If every member of A is also a member of B, then A is the subset of B, shown by If every member of A is also a member of B, then A is the subset of B, shown by We can also write it as B is the superset of A, shown by We can also write it as B is the superset of A, shown by is called inclusion or containment is called inclusion or containment A is a subset of X B is a subset of X

Proper Subsets However, if members of A is a subset, but NOT equal to B, then A is called a proper subset of B, written (or A is a proper superset of B, written ). However, if members of A is a subset, but NOT equal to B, then A is called a proper subset of B, written (or A is a proper superset of B, written ). can also be written as can also be written as

Special Sets All of these sets are represented using Blackboard bold typeface All of these sets are represented using Blackboard bold typeface denotes the set of all primes denotes the set of all primes denotes the set of all natural numbers denotes the set of all natural numbers denotes the set of all integers (positive, negative and zero) denotes the set of all integers (positive, negative and zero) denotes the set of all rational numbers denotes the set of all rational numbers denotes the set of all real numbers denotes the set of all real numbers denotes the set of all complex numbers denotes the set of all complex numbers Although each of these sets have infinite size, the order of the special sets is although the primes are used less Although each of these sets have infinite size, the order of the special sets is although the primes are used less

Union Sets can be “added” together The union of two sets is the set of elements that are in at least one of the two sets. For example, if A={1, 2, 3, 4} and B={2, 4, 6, 8} then A B = {1, 2, 3, 4, 6, 8}. then A B = {1, 2, 3, 4, 6, 8}. “the union of A and B contains {1,2,3,4,6,8}

Intersection The intersection of two sets is the elements they have in common. For example, if A={1, 2, 3, 4} and B={2, 4, 6, 8}, then A B = {2, 4}. “the intersection between A and B contains the numbers {2,4}

Complement A` means everything except A. You can see it from the diagram below A` = X - A

Example If there are 50 people in the class, 20 wears glasses and 24 wears watches and 15 wears both, how many people do not wear anything? X = {{G,W} – {B}}+{N} G and W is a subset of X X = everything within that range = 50 = 50 (50) = (24) + (20) – (15) + N = 29 + N N = 21 = 29 + N N = 21 GW B B

Challenge Questions There are 100 students in Year 10. In athletics day there are: 52 people running 100 meters 52 people running 100 meters 46 people running 200 meters 46 people running 200 meters 33 people running 400 meters 33 people running 400 meters 20 people running both 100 and people running both 100 and people running both 200 and people running both 200 and people running ONLY people running ONLY people running ONLY people running ONLY 100 How many students are absent on that day?

The solution 52 people running 100 meters 52 people running 100 meters 46 people running 200 meters 46 people running 200 meters 33 people running 400 meters 33 people running 400 meters 20 people running both 100 and people running both 100 and people running both 200 and people running both 200 and people running ONLY people running ONLY people running ONLY people running ONLY 100

More challenge questions There are 66 students and 19 teachers in ABC elementary school. There are: 36 boys 36 boys 30 girls 30 girls 19 teachers 19 teachers 59 people wearing watches 59 people wearing watches 56 people wearing glasses 56 people wearing glasses The same number of boys and girls wearing glasses The same number of boys and girls wearing glasses The number of teacher wearing glasses is 1 less than the number of boys wearing glasses The number of teacher wearing glasses is 1 less than the number of boys wearing glasses 38 people wearing both 38 people wearing both The number of girls wearing watches is 5 less than the number of boys wearing watches The number of girls wearing watches is 5 less than the number of boys wearing watches 20 girls wearing watches 20 girls wearing watches There are 20 boys wearing only one object There are 17 girls wearing only one object There are 17 girls wearing only one object How many students do not wear neither watch or glasses?

Venn diagrams Label/ shade these A’ Label/ shade these A’ A B

Venn diagrams Label/ shade these B’ Label/ shade these B’ A B

Venn diagrams Label/ shade these A B Label/ shade these A B A B

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Venn diagrams Label/ shade these A’ B Label/ shade these A’ B A B

Venn diagrams Label/ shade these A’ B Label/ shade these A’ B A B