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Sets and Counting. JA Ritcey EE 416 Revised Oct 5 2009 Please elaborate with your own illustrations!. Elementary Set Theory. Universe – superset containing any element of interest. This is the complement of the empty set. Subset – collection of none, some, or all elements - PowerPoint PPT Presentation
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Sets and Counting
JA RitceyEE 416
Revised Oct 5 2009
Please elaborate with your own illustrations!
Elementary Set Theory
• Universe – superset containing any element of interest. This is the complement of the empty set.
• Subset – collection of none, some, or all elements• Set operations – complement, union, intersection• Cardinality – the number of elements in a set, size• Cardinality can be finite, countable, or uncountable
• Venn diagrams are useful pictorial representations• Set operations can be carried out in MATLAB!
Set Operations
• U = { u_1, … , u_N } with card(U) = N• Count by placing elements of U in one-to-one
correspondence with the natural numbers
• You should be able to carry out set operations by taking unions, intersections, and complements.
• You should be able to handle both lists and geometric regions.
Disjoint or Mutually Exclusive Sets
• 2 sets a disjoint or mutually exclusive if they have no common elements
• disjoint means the intersection is the empty set
• Disjoint sets are simpler than those with overlap
• A collection of sets is mutually disjoint if no two intersect. Together they partition the union.
• Partition – a collection of disjoint subsets that is complete and together yields the entirety
Partitions
• A set S is partitioned by a collection of disjoint subsets that together encompass all of S
• The days of the week are partitioned into work days and weekend
• The numbers {1,2,3, …} are partitioned into odd and even
• Partitions are used to when something is held invariant over each subset
Sensor Nodes Partitioned by Radius from Acces Point
Cartesian Products
• Given 2 sets A = { a_i } B = { b_j } create the • Ordered pair c = (a_i, b_j )
• C = AxB is called the Cartesian product• |C| = |AxB| = |A|x|B| . This is called the
multplication principle in counting (combinatorics)
• Select one from column A and 1 from column B
• This can be extended to ordered J-tuples• C= A_1 x … x A_J the J-fold product• A vector can be created this way
Power Sets
• Power Set Pwr(S) is the set of all subsets of any S
• If a set S is finite, say |S| = N, then• |Pwr(S)| = 2^N• To see this equate each member of Pwr(S) with
the binary string (b_1, b_2, … , b_N ).• The b_i in (0,1) label whether element s_i is in
the particular subset.• There are 2^N binary strings of length N
• If the cardinality, or size, of A is countable, the power set has uncountable cardinality
Properties of the Power Set• Pwr(S) has some closure properties• A collection of sets C is closed under set operations:• (1) If A in C, A^c in C (closed under complements)• (2) If A_1, …A_N in C, their union is in C
• Often this last property is extended to countable unions. If A_1, A_2,… in C, their union is in C
• Collections of sets with this property are called algebras and sigma-algebras.
• When S is uncountable Pwr(S) is too big to assign probabilities to everything – a subtle point
Generating some small power sets
• function powerset(n);• %function
powerset(n);
• n1=n-1;• for knt = 0:2^n1-1• C= bitget(knt,n1:-
1:1);• disp([knt C])• end
0 0 0 0 0 1 0 0 0 1 2 0 0 1 0 3 0 0 1 1 4 0 1 0 0 5 0 1 0 1 6 0 1 1 0 7 0 1 1 1 8 1 0 0 0 9 1 0 0 1 10 1 0 1 0 11 1 0 1 1 12 1 1 0 0 13 1 1 0 1 14 1 1 1 0 15 1 1 1 1
Counting Finite Sets
• We have already done some counting, using the Multiplications Principle
• Cartesian product card |C| = |AxB| = |A| |B|• All pairs must be possible• The size of A and B can be different, but finite
• There are 52 = 4x13 cards in a deck• There are 100 = 2x50 US Senators
• Lets call a set S an N-set when |S|=N
Ordered Selection w/ Repetition
• Given a base N-set S• In how many ways can we select k objects• Assume an ordered selected With
replacement. So that repetition is possible
• We can do this is N^k ways by the multiplication principle
• Just equate the selection of an object with an ordered k-tuple The elements are independently selected and each drawn from an N-set (repetitions)
Tabular View
• Repetition• With Without• • W/ Order N^k N!/(N-k)!
• WO/ Order (N+k-1)!/ (N-1)! k! N!/(N-k)!k!• (N+k-1 choose k) (N choose
k)
• It is easier to remember these, when they are considered as a group
Ordered Selection W/Replacement
• Each Selection is a row• Possible Choices along
columns
• 7^12 = 7 x7 …x7
• Cartesian Product
• Illustrates the Multiplication Principle
1 2 3 4 5 6 7
2
3
4
5
6
7
8
9
10
11
12
Examples
• Cards = 4 suits x 13 face values = 52 cards
• You can place r balls into n cells in n^r ways• n^r = n x n … x n
• Harder: How many ways can r flags be put on n flagpoles?
• Ans: n x (n+1) x (n+2) x … x (n+r-1)• Why: The second flag can be put above or
below the first. So the number of choices increases
Ordered Selection wo/ Repetition
• Given a base N-set S• In how many ways can we select k objects• Assume an ordered selected without
replacement. So that repetition is not possible• We can do this is (N)_k = N(N-1) …(N-k+1)
ways, again by the multiplication principle• Just equate the selection of an object with an
ordered k-tuple The elements are each drawn from sets of decreasing size
• When k=N, we can select in N! ways
Tabular View
• Repetition• With Without• • W/ Order N^k N!/(N-k)!
• WO/ Order (N+k-1)!/ (N-1)! k! N!/(N-k)!k!• (N+k-1 choose k) (N choose
k)
• It is easier to remember these, when they are considered as a group
Factorial Interpretation
• The factorial n! = n(n-1)(n-2) …(3)(2)(1) 0!=1• It gets very big very fast,• n! asy sqrt(2pi)n^(n+1/2)exp(-n)
• Then (N)(N-1)…(N-k+1) = N! / (N-k)!
• Best ways to compute the factorial is through the Gamma Function (gamma.m) More later.
• You can permute (reorder) an N-tuple in N! ways
Factorial Growth
Stirling’s Formula
Matlab M-File• functioncompareGamma(M)
;
• m = [1:M]';half=1/2;• fact = gamma(m+1);• plot( m, fact, 'b*-');grid• title('m! = Gamma(m+1) -
Grows very fast');• xlabel('m');ylabel('G(m)');
• %try logs to compress the dynamic range
• figure(2)• subplot(121);• stir =• sqrt(2*pi).*m.^(m+half).*exp(-
m);• semilogy( m, fact,
'b*-',m,stir,'ro');grid• subplot(122);• RE = abs(stir-fact)./fact;• semilogy(m,RE,'r*');grid• title('Relative Error');• figure(1)
Unordered selection wo/ Repetition
• What if we select without regard to the order.• Solve this by reconsidering ordered selection
wo/ replacement of k objects from N-set• Each such selection resulted a k-tuple that
indexed the selection a_i is that object selected on the i^th turn
• But a k-tuple can be reordered (permuted) in k! ways
• Therefore we need to reduce our result to• N(N-1)…(N-k+1)/k! = N! / (N-k)! K! • Combinations - binomial coefficient ( N choose
k )
Tabular View
• Repetition• With Without• • W/ Order N^k N!/(N-k)!
• WO/ Order (N+k-1)!/ (N-1)! k! N!/(N-k)!k!• (N+k-1 choose k) (N choose
k)
• It is easier to remember these, when they are considered as a group
Computing Binomial Coefficient
• { n choose k } = n! / (n-k)! k!• But n! = gamma(n+1) = G(n+1) (gamma.m)
• { n choose k } = G(n+1)/G(n+1-k) G(k+1)
• Even better to use loggamma = log(Gamma(m))
• { n choose k } = • exp( gammln(n+1) -gammaln(n+1-k) -
gammaln(k+1) )
More Binomial Coefficient
• binomialcoef(n) Think (1+x)^n. The coefficients:
• 1 1• 1 2 1• 1 3 3 1• 1 3 5 3 1• 1 4 9 9 4 1• 1 6 15 19 15 6 1
• Computation – recursion, approximate, logGamma
Matlab Code
• function binomialcoef(n);• %generates an array of binomial coefficients
• k = [0:n];• lgGn = gammaln(n+1);• lgGnk = gammaln(n+1-k);• lgGk = gammaln(k+1);• • bico = exp( lgGn -lgGnk -lgGk );• bico = floor(bico);• fprintf('%5.0f', bico);
Unordered w/ Repetition
• This is the most complicated situation so far• Thinks for a long time, then come up with a• Clever way to represent a selection bars & balls• 4 objects choose 2 o||o| select object 1 and 3• Another choice is |||oo select object 4 twice
• To select k objects from N-set • Use N-1 bars and k balls string of |||oo• But these can be drawn in ( N+k-1 choose k)
ways• (N+k-1)!/(N-1)!k!
Tabular View
• Repetition• With Without• • W/ Order N^k N!/(N-k)!
• WO/ Order (N+k-1)!/ (N-1)! k! N!/(N-k)!k!• (N+k-1 choose k) (N choose
k)
• It is easier to remember these, when they are considered as a group
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