Transcript Document

Basics of Set Theory
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Set notation: {1, 2, 3}, {{1, 2}, {3}, {1, 2, 3}}, {1, 2, 3,
…}, , {x  R | -3 < x < 6}.
Set A is called a subset of Set B, written as A  B,
when x, x  A  x  B.
A is a proper subset of B, when A is a subset of B and
x  B and x  A.
Visual representation sets: Venn diagrams.
Note the distinction between  (set containment) and
 (set membership).
Set Operations
Let A and B be subsets of a universal set U.
 Union of two sets
A  B = {x  U | x  A or x  B }
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Intersection of two sets
A  B = {x  U | x  A and x  B }
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Difference of two sets
B ─ A = {x  U | x  B and x  A }
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Complement of a set
Ac = {x  U | x  A }
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Equality of two sets
A = B  A  B and B  A
Cartesian Products
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The Ordered n-tuple, (x1, x2, …, xn), consists of elements
x1, x2, …, xn ordered positionally.
 Equality of n-tuples?
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The Cartesian product of n sets, A1 x A2 x …x An, is a set
of n-tuples, where each element in the n-tuple belongs to
the respective set participating in the product.
Power Set
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The power set of A, denoted P (A), is the set of
all subsets of A.
Theorem: If A  B, then P (A)  P (B).
Theorem: If set A has n elements, then P (A)
has 2n elements.
Set Partitioning
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Two sets are called disjoint if they have no elements
in common.
Theorem: A – B and B are disjoint.
A collection of sets A1, A2, …, An is called mutually
disjoint when any pair of sets from this collection is
disjoint.
A collection of non-empty sets {A1, A2, …, An} is
called a partition of a set A when the union of these
sets is A and this collection consists of mutually
disjoint sets.
More on Empty Set
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S = {x  R | x2 = -1}.
X = {1, 3}, Y = {2, 4}, C = X  Y.
Empty set  has no element.
Empty set is a subset of any set.
Theorem: There is exactly one empty set.
Properties of empty set:
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A   = A, A   = 
A  Ac = , A  Ac = U
Uc = , c = U
More on Set Properties
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Inclusion of Intersection:
 A  B  A and A  B  B
Inclusion in Union:
 A  A  B and B  A  B
Transitivity of Inclusion:
 (A  B  B  C)  A  C
Set Definitions:
Let X, Y be subsets of a universal set U and x, y be elements of U.
 xXYxXxY
 xXYxXxY
 xX–YxX xY
 x  Xc  x  X
 (x, y)  X  Y  x  X  y  Y
Set Identities
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Commutative Laws: A  B = B  A and A  B = B  A
Associative Laws: (A  B)  C = A  (B  C) and (A  B)  C = A  (B  C)
Distributive Laws:
A  (B  C) = (A  B)  (A  C) and A  (B  C) = (A  B)  (A  C)
Intersection and Union with universal set: A  U = A and A  U = U
Double Complement Law: (Ac)c = A
Idempotent Laws: A  A = A and A  A = A
De Morgan’s Laws: (A  B)c = Ac  Bc and (A  B)c = Ac  Bc
Absorption Laws: A  (A  B) = A and A  (A  B) = A
Alternate Representation for Difference: A – B = A  Bc
Intersection and Union with a subset: if A  B, then A  B = A and A  B = B
Subset Check Algorithm
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Let two sets be represented as arrays A and B
m = size of A, n = size of B
i = 1, answer = “yes”;
while (i  m && answer == “yes”) {
j = 1, found = “no”;
while (j  n && found == “no”) {
if (a[i] == b[j]) found = “yes”;
j++;
}
if (found == “no”) answer = “no”;
i++;
}
Exercises
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Is is true that (A – B)  (B – C) = A – C?
Show that (A  B) – C = (A – C)  (B – C)
Is it true that A – (B – C) = (A – B) – C?
Is it true that (A – B)  (A  B) = A?
Exercises
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Simplify: A  ((B  Ac)  Bc) (P291, Q.34)
Symmetric Difference: A  B = (A – B)  (B – A)
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Show that symmetric difference is associative. (P292, Q.45)
Are A – B and B – C necessarily disjoint? (P291, Q.33)
Are A – B and C – B necessarily disjoint? (P291,Q.7)
Let S = {2, 3, …, n}. For each nonempty Si  S, let
Pi be the product of elements in Si. Show that:
Pi = (n + 1)! / 2 – 1 (P291, Q.22)
Boolean Algebra
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Boolean Algebra is a set B together with two
operations denoted as + and *, such that for all
a and b in B both a+b and a*b are in B and the
following properties hold:
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a + b = b + a, a * b = b * a
(a + b) + c = a + (b + c), (a * b) *c = a * (b * c)
a + (b * c) = (a + b) * (a + c), a * (b + c) = (a * b) + (a * c)
a + 0 = a, a * 1 = a (for distinct and unique elements 0 and 1)
a + ã = 1, a * ã = 0 (ã is the complement of a)