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Discrete Mathematics Ch. 5 Sets (Review)

Discrete Mathematics Ch. 5 Sets (Review). Today we will review sections 5.1, 5.2, 5.3. Instructor: Hayk Melikyan melikyan@nccu.edu. What is a set? A collection of elements: Order is irrelevant No repetitions Can be infinite Can be empty Examples: {Angela, Belinda, Jean}

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Discrete Mathematics Ch. 5 Sets (Review)

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  1. Discrete MathematicsCh. 5 Sets (Review) Today we will review sections 5.1, 5.2, 5.3 Instructor: Hayk Melikyan melikyan@nccu.edu

  2. What is a set? • A collection of elements: • Order is irrelevant • No repetitions • Can be infinite • Can be empty • Examples: • {Angela, Belinda, Jean} • {0,1,2,3,…}

  3. Operations on sets • S is a set Membership: xS x is an element of S Angela{Angela, Belinda, Jean} • Subset S1  S • Set S1is a subset of set S • All elements of S1 are elements of S • {Angela,Belinda}  {Angela, Belinda, Jean} • Proper subset S1 S

  4. Operations on sets • If S, S1 are sets • Intersection: S  S1 • is a set of all elements that belong to both {Ang, Bel, Jea}  {Ang, Dan} = {Ang} • Union: S  S1 • is a set of all elements that belong to either • {Ang, Bel, Jea}  {Ang, Dan} = {Ang, Bel, Jea, Dan}

  5. Operations on sets • Let S, S1 be sets • Equality: S = S1 • iff they have the same elements • Difference: S \ S1 • is a set of all elements that belong to S but NOT to S1 {Ang, Bel, Jea} \ {Ang, Dan} = {Bel, Jea}

  6. More notation • In mathematics sets are often specified with a predicate and an enveloping set as follows: S = {xA | P(x)} S is the set of all elements of A that satisfy predicate P • Example: Q={xR | a,bZ b0 & x=a/b}

  7. Set Equality • Two sets are equal iff they have the same elements • Theorem: for any sets A and B, A=B iff AB & BA

  8. Book example 5.1.5 • 2{1,2,3} ? • {2}{1,2,3} ? • 2{1,2,3} ? • {2}{1,2,3} ? • {2}{{1},{2}} ? • {2}{{1},{2}} ? • How about set A such that {2} is a subset of it and A is an element of it? • A={1,2,{1},{2}}

  9. Universal Set • If we are dealing with sets which are all subsets of • a larger set U then we call it a universal set U • All of your sets will be subsets of U • When does such a U exist? Always, for we can set U to the union of all sets involved ???????

  10. Complement • So if I am dealing with set A which is a subset of the universal set U then: • I can define complement of A: AC = U\A • That is the set of all elements (of U) that are not in A • Often “of U” is dropped and people say that ACis • the set of everything that is not in A What is the complement of U? UC = Ø What set has U as its complement? ØC=U

  11. Sets & Predicate Logic • All of the set operations and relations above can be defined in terms of Boolean connectives: • AB={x | xA v xB} • AB={x | xA & xB} • A\B={x | xA & not xB} • AC={x | not xA} • A = B iff x( xA  xB) • A  B iff x (xA  xB) • AB iff x (xA  xB) & not A=B

  12. Symmetric Difference • Set C is the symmetric difference of sets A and • B iff every element of C belongs to AorB but • not both • ABC [C=A  B  a (aC  (aA xor aB))] If A={1,2}, B={2,3} then A  B={1,3} In general: A  A = {}

  13. Exercise 2 • Intersection of two sets is contained in their union: AB [ (A  B)  (A  B) ] • Proof:

  14. Exercise 3 • Union is commutative AB [ A  B = B  A ] Intersection is commutative AB [ A  B = B  A ] Intersection distributes over union: ABC [ A  (B  C) =(A  B)  (A  C) ]

  15. Exercise • In Exercise #5 we proved: • ABC [ A  (B  C) =(A  B)  (A  C) ] • using the fact that A&(BvC) = (A&B)v(A&C) • Given the statement just proved Av(B&C) = (AvB) & (AvC) • what can we now prove in terms of sets? • Union distributes over intersection: ABC [ A  (B  C) =(A  B)  (A  C) ]

  16. Proof :

  17. Logic - Sets • v • & • avb=bva ab=ba • a&b=b&aab=ba • (avb)vc=av(bvc) (ab)c=a(bc) • (a&b)&c=a&(b&c)(ab)c=a(bc) • a&(bvc)=(a&b)v(a&c)a(bc)=(ab)(ac) • av(b&c)=(avb) & (avc)a(bc)=(ab)(ac)

  18. Cartesian Products • Intuition first: • Suppose I have a function that takes two numbers x and y and returns x/y • What is the set of valid inputs? • Is it just R? • No -- cannot divide by 0 • Is it R\{0}? • No -- can happily have 0 as x

  19. Combinations • Suppose I have: • two independent attributes: sky conditions and temperature two values for the sky conditions S={sunny, overcast}; three values for the precipitation: P={snow, rain, nothing}. • How many combinations can I have? • <sunny, rain> , <sunny, snow> . <sunny, nothing>. • <overcast, rain>. <overcast, snow>, <overcast, nothing>

  20. Cartesian Product • Set C is a Cartesian Product of set A and set B iff it is a set of all ordered pairs such that the 1st element belongs to A and the 2nd element belongs to B • C=A  B iff ab (<a, b>C  (aA & bB))]

  21. Examples • A={0,1}, B={Ang, Bel} • AB = {<0,Ang>, <0,Bel>, <1,Ang>, <1, Bel>} • A={0,1}, B={Ang, Bel} • BA = {<Ang, 0>, <Bel, 0>, <Ang, 1>, <Bel,1>} • A={0}, B={a,b}, C={1,2} • ABC={<0, a, 1>,<0, b, 1>, <0, a, 2>,<0, b, 2>}

  22. More examples • A=B=C=D=R (set of all real numbers) • ABCD=R4 (time-space continuum) • What is the cardinality of Cartesian Product? • |A1  …  An|=|A1| · … · |An| for finite sets • How about { }  {1,2}? • {}  {1,2}={}

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