Pamela Leutwyler

definition

a set is a collection of things – a set of dishes a set of clothing a set of chess pieces

In mathematics, the word “set” refers to “a well defined collection.” The “collection of all short people” is not well defined (not a set) The “collection of all people less than 4 feet tall” is well defined (a set)

notation

abcde this collection of letters forms a set

abcde separate the members of the set with commas,,,, surround the members of the set with braces {}

abcde,,,, {} It is conventional to use an upper case letter to name a set. S =

abcde,,,, {}  this symbol means “ is a member of ”

abcde,,,, {} S =  this symbol means “ is a member of ” “d is a member of the set S” “d S”  is written

abcde,,,, {} S =  this symbol means “ is a member of ” not

abcde,,,, {} S =  this symbol means “ is a member of ” “h is not a member of the set S” “h S”  is written not

abcde,,,, {} S = bce,, {} R =

abcde,,,, {} S = bce,, {} R = the set R is related to the set S

abcde,,,, {} S = bce,, {} R = the set R is related to the set S every element of R is also an element of S

abcde,,,, {} S = bce,, {} R = R is called a SUBSET of S

abcde,,,, {} S = bce,, {} R = R is called a SUBSET of S denoted R  S

the set of CANARIES is a subset of the set of BIRDS

the set of WOMEN is a subset of the set of HUMANS

the set of all COUNTING NUMBERS is called the set of NATURAL NUMBERS and is denoted “N” N = { 1, 2, 3, 4, 5, 6, 7, 8, … }

the set of all COUNTING NUMBERS is called the set of NATURAL NUMBERS and is denoted “N” N = { 1, 2, 3, 4, 5, 6, 7, 8, … } the set of EVEN counting numbers is a subset of N E  N

two sets are said to be EQUAL if and only if they contain the same members. {1, 2, 3 } = { 2, 1, 3 }

in arithmetic we use the symbols: , , ,  greater than greater than OR equal to less than less than OR equal to If your age is less than 16, you cannot drive legally a  16 If your age is greater than or equal to 16, you can drive legally a  16

We use similar notation in set theory. Suppose we know that every member of a set A is also a member of a set C.

We use similar notation in set theory. Suppose we know that every member of a set A is also a member of a set C. If it is possible that A is equal to C, we write A  C example: A = the set of people enrolled in this class who will get an A in this class C = the set of people enrolled in this class

A  C B  D If you know that every member of A is also a member of C then this symbol is always correct If you know that every member of B is also a member of D and you also know that there is at least one member of D that is not in B, you can use this symbol. In this case, B is called a PROPER SUBSET of D.

a set that has no members is called “EMPTY” and is represented with this symbol:  example: the set of all elephants who can sing opera in French = 

the set that contains everything (or everything that you are talking about) is called the “UNIVERSE” and is represented by the symbol: U if you are solving algebraic equations then U = the set of real numbers if you are a doctor studying the correlations of smoking with heart disease then U = the set of people in your sample population for any discussion of sets, the universe must be defined.

Sometimes it is difficult or impossible to list all of the members of a set. We can use words to describe such sets: the set of all people enrolled in this class the set of all counting numbers that are greater than 5 the set of all members of the US senate

“set builder notation” is the most efficient and accurate way to describe these sets. A variable symbol is used to designate an unspecified member of the Universe: the set of all people enrolled in this class the set of all counting numbers that are greater than 5 the set of all members of the US senate = { x

“set builder notation” is the most efficient and accurate way to describe these sets. A variable symbol is used to designate an unspecified member of the Universe: this slash is usually read “such that” the set of all people enrolled in this class the set of all counting numbers that are greater than 5 the set of all members of the US senate = { x /

“set builder notation” is the most efficient and accurate way to describe these sets. A variable symbol is used to designate an unspecified member of the Universe: this slash is usually read “such that” the sentence gives the condition that defines membership. the set of all people enrolled in this class the set of all counting numbers that are greater than 5 the set of all members of the US senate = { x / x is a person enrolled in this class}

“set builder notation” is the most efficient and accurate way to describe these sets. A variable symbol is used to designate an unspecified member of the Universe: this slash is usually read “such that” the sentence gives the condition that defines membership. the set of all people enrolled in this class the set of all counting numbers that are greater than 5 the set of all members of the US senate = { x / x is a person enrolled in this class} = { x / x is a US senator } = { x / x  N and x  5 }

examples: { x / x  N and 5  x  9 } = { 6, 7,8, 9 }

examples: { x / x  N and 5  x  9 } = { 6, 7,8, 9 } { 2x / x  N and 5  x  9 } = { 12, 14,16, 18 }

the number of members in a set S is called the “cardinal number of S” denotes “n(S)” A = { 1, 2, 3, 4 } n(A) = 4 V = { a, e, I, o, u } n(V) = 5 H = { p, q, r, s } n(H) = 4

the number of members in a set S is called the “cardinal number of S” denotes “n(S)” A = { 1, 2, 3, 4 } n(A) = 4 V = { a, e, I, o, u } n(V) = 5 H = { p, q, r, s } n(H) = 4 Two sets have the same cardinal number if and only if it is possible to establish a one to one correspondence between their members. Two sets that have the same cardinal number are said to be EQUIVALENT.

E = the set of even counting numbers. n( E ) = n( N ) E = { 2, 4, 6, 8,……………….. 2n,………} N = { 1, 2, 3, 4, 5, 6, 7, 8, ……,n,……….}

operations

suppose: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = { 1, 2, 3, 4, 5, 6 } B = { 2, 4, 6, 8, 10 }

suppose: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = { 1, 2, 3, 4, 5, 6 } B = { 2, 4, 6, 8, 10 } { x / x  A } =

suppose: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = { 1, 2, 3, 4, 5, 6 } B = { 2, 4, 6, 8, 10 } { x / x  A } = { 7, 8, 9, 10 } this set is called the “complement” of A and is denoted “ A’ ” A’ =

suppose: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = { 1, 2, 3, 4, 5, 6 } B = { 2, 4, 6, 8, 10 } { x / x  A AND x  B } =

suppose: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = { 1, 2, 3, 4, 5, 6 } B = { 2, 4, 6, 8, 10 } { x / x  A AND x  B } = { 2, 4, 6 } this set is called the “intersection” of A with B and is denoted “A  B” A  B =

suppose: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = { 1, 2, 3, 4, 5, 6 } B = { 2, 4, 6, 8, 10 } { x / x  A OR x  B } =

suppose: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = { 1, 2, 3, 4, 5, 6 } B = { 2, 4, 6, 8, 10 } { x / x  A OR x  B } = { }

suppose: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = { 1, 2, 3, 4, 5, 6 } B = { 2, 4, 6, 8, 10 } { x / x  A OR x  B } = { 1, 2, 3, 4, 5, 6, 8, 10 } this set is called the “union” of A with B and is denoted A  B A  B =

suppose: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = { 1, 2, 3, 4, 5, 6 } B = { 2, 4, 6, 8, 10 } { x / x  A and x  B } =

suppose: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = { 1, 2, 3, 4, 5, 6 } B = { 2, 4, 6, 8, 10 } { x / x  A and x  B } = { 1, 3, 5 } The members of B are removed from A. This is what remains.

suppose: U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } A = { 1, 2, 3, 4, 5, 6 } B = { 2, 4, 6, 8, 10 } { x / x  A and x  B } = { 1, 3, 5 } this set is called “A minus B” and is denoted “A – B” A - B =