Alg Concept of Sets

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    Concept of Sets

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    2

    REMEMBER THIS.

    DO NOT MEMORIZE.

    UNDERSTAND!

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    y If you think mathematics is difficult and boring, you are right. It is difficult and boring.

    y If you think it is easy and fascinating, you are also right. It is easy and fascinating.

    y It is all in the mind!

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    Basic DefinitionsSet - a collection of elements or objects of interest

    Em pty set (denoted by )a set cont ai n ing no ele m ents

    E lem ent or m em ber of a setobject th a t belongs to a set, denoted by Un iv ersa l set (denoted by S)

    a set cont ai n ing a ll poss ible ele m entsCom plem ent ( Not). The co m plem ent of A is

    a set cont ai n ing a ll elem ents of S not in A

    4

    A

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    A set set , informally, is a collection of things. The "things" in the setare called the " elementselements", and are listed inside braces. For instance, to list the elements of "the set of things on a kids bed,"the set, denoted by A A,, would look like this:

    A = {pillow, rumpled bedspread, a stuffed animal, one cat taking a nap}

    Sets are "unordered", which means that the things in the set donot have to be listed in any particular order.

    To say that something is an element of a set, for instance, that a"pillow is an element of the set A ", we would write the following :

    This is pronounced "pillow is an element of A ".

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    The elements of a set can be listed out according to a rule, such as:

    { x is a natural number, x < 10}

    You can use full "set-builder notation", which looks like this:

    pronounced as "the set of all x , such that x is an elementof the natural numbers and x is less than 10".

    The vertical bar is usually pronounced as "such that", and it comesbetween the name of the variable used to stand for the elementsand the rule that says what those elements actually are.

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    This same set, since the elements are few, can also be given by alisting of the elements, like this:

    {1, 2, 3, 4, 5, 6, 7, 8, 9}

    Listing the elements explicitly like this, instead of using a rule,is often called " u sing the roster method ".

    Sets can be related to each other. If one set is "inside" another set, itis called a "subset". Suppose A = {1, 2, 3} and B = {1, 2, 3, 4, 5, 6}.Then A is a subset of B , since everything in A is also in B .

    This is written as:

    That sideways-U thing is the subset symbol, and is pronounced"is a subset of".

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    To show something is not a subset, you draw a slash through thesubset symbol, so the following:

    ...is pronounced as " B is not a subset of A ".

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    Venn diagramy V enn diagrams , Euler Euler diagramsdiagrams (pronounced "o iler")

    a nd Johnston diagrams Johnston diagrams a re s im ila r-look ing illustr a t ionsof sets , mathematical mathematical or logical logical rela t ionsh ips.

    y Ex ample

    y The Venn d ia gra m a bo v e ca n be interpreted a s "therela t ionsh ips of set A a nd set B which m a y hav e som e(but not a ll) elem ents in comm on".

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    The Euler diagram above can be interpreted as "set A is a proper subsetof set B , but set C has no elements in common with set B .

    Or, as a syllogismsyllogism

    All V s are T s All K s are V sTherefore All K s are T s.

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    C omplement of a Set

    A

    A

    S

    11

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    I ntersection (And)a set cont ai n ing a ll elem ents in both A a ndB

    U nion (Or)a set cont ai n ing a ll elem ents in A or B orboth

    Basic Definitions ( C ontinued)

    12

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    Sets: A Intersecting with B

    A

    B

    S

    13

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    Sets: A Union B

    A

    B

    S

    14

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    If two sets are being combined, this is called the " u nion" of thesets, and is indicated by a large U-type character. If instead of taking everything from the two sets, you're only taking what iscommon to the two, this is called the " intersection" of the sets,and is indicated with an upside-down U-type character. So if C = {1, 2, 3, 4, 5, 6} and D = {4, 5, 6, 7, 8, 9}, then :

    These are pronounced as " C union D equals..." and " C intersectD equals...", respectively.

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    Lets Seey Give a solution using the roster method:

    A= { 1, 2, 3, 4, 5, 6, 7 }, B is a subset of A, the elements of B areeven.The numbers in A that are even are 2, 4, and 6, so B = {2, 4, 6} .

    y W hat is the intersection of A = { x is odd } and B = { x is between 4 and 6 }, where the elements of the two sets are integers?

    Since "intersection" means "only things that are in both sets", theintersection will be all the numbers which are both odd and between

    4 and 6. {3, 1, 1, 3, 5}

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    Lets See

    y W hat is the union of A = { x is a natural number between 4 and 8inclusive } and

    B= {

    x is a single-digit negative integer }?

    Since "union" means "anything that is in either set", the union will beeverything from A plus everything in B . Since A = { 4, 5, 6, 7, 8 } and B = {

    9, 8, 7, 6, 5, 4, 3, 2, 1 }, then their union is:{ 9, 8, 7, 6, 5, 4, 3, 2, 1, 4, 5, 6, 7, 8 }

    y Give a solution using a rule: The set of all the odd integers.

    An odd integer is one more than an even integer, and every even integer is a multiple of 2, so the set would be:

    { x Z x = 2m + 1, m Z}

    pronounced as "all integers x such that x is equal to 2 times m plus 1,where m is an integer", which is a fancy way of saying " x is an oddinteger". It's a lot easier to describe this set using the roster method: {..., 3, 1, 1, 3, 5, 7, ...}. The ellipsis (the three periods in a row) means "and soforth", and indicates that the pattern continues indefinitely in the givendirection.

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    Mu t u ally e x clu sive or disjoint setssets h avi ng no ele m ents in comm on, h avi ng

    no intersect ion, whose intersect ion is theem pty setPartition

    a collect ion of m utu a lly exclusiv e sets which together include a ll poss ibleelem ents, whose un ion is the un iv ersa l set

    Basic Definitions ( C ontinued)

    18

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    M utually Exclusive or Disjoint Sets

    AB

    S

    Sets have nothing in commonSets have nothing in common

    19

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    Sets: Partition

    A1

    A2

    A3

    A4

    A5

    S

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    Sets: P(A Union B)

    A

    B

    S

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    K inds of Setsy E mpty/N u ll/Void Set a se t tha t has no ele m e nt s, de not ed

    by { } or E x. T he se t of nu m be r s i n the E ng lish alp habe t .

    Finite Set - a se t wi th c ount able nu m be r of ele

    me nt sE x. T he se t of le tt e r s i n the E ng lish

    alp habe t .

    Infinite Set- a se t tha t has un c ount able nu m be r of ele m e nt s .E x. T he se t of c ount i ng nu m be r s .

    Universal Set - all the ele m e nt s of the se t un de r c on side r a t i on ( U)E x. T he se t of r eal nu m be r s .

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    R ela t ionsh ip of Setsy Eq ual Sets - sets with the same elements.

    Ex. A = {t.a.m.e};B = {m.a t,e}. A = B

    y

    Eq uivalent Sets - sets with the same number of elements.Ex. Let C = {x x is nei the r posi t ive nor ne g a t ivei nt e g e r) a nd le t D = { x x is a n eve n pr i m e

    nu m be r }.S i nce C = { 0 } a nd D = { 2 }, w he r ei n both C a nd D have on ly on e ele m e nt; the n C is eq u ivale nt to D , de not ed by C ~ D.

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    y J oint Sets sets with at least one common element.Ex. Let E = {x x is the se t of posi t ive f ac tor s of 6}a nd le t F = { x x is the se t of posi t ive f ac tor s of 9}. S i nce E = { 1,2,3,6 } a nd F = { 1,3,9 }, the n thesese t s a r e se t to be j o i nt se t s .

    y D isjoint Sets sets that have no common element.Ex. Let G = { x x is the se t of eve n nu m be r s } a nd

    Le t H = { x x is the se t of odd nu m be r s } .T he r e for e, se t s G a nd H a r e disj o i nt se t s si nce no eve n nu m be r is a n odd nu m be r a nd vice ve r sa .

    R elationship of Sets..

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    Relationship

    y

    Subset set every element of which can be on the second set. The symbol means a subset of. If the first set equals the second set, then it is an improper subset.

    Ex. Let P = { x x is the las t 3 le tt e r s i n the E ng lish alp habe t = { x , y z } . S i nce P has 3 ele m e nt s the n the p owe r se t has 2 3 = 8 ele m e nt s . Le t S be the

    p owe r se t thu s,

    S = { {

    x }, {y}, {Z}, {

    x , y}, {

    x , z}, {y, z}, P, }S e t s { x }, {y}, {Z}, { x , y}, { x ,Z}, {y, z} a r e p ro pe r s ubse t s of

    P w h ile the se t P i t sel f a nd a r e i t s i m p ro pe r s ubse t s .