SETS
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Transcript of SETS
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SETS
A set B is a collection of objects such that for every object X in the universe the statement:
“X is a member of B”
Is a proposition.
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A quick review of basic notation and set operations.
1. A = {1, 2, ab, ba, 3, moshe, table},
2.1,2, ab, ba, moshe table are “elements.” They are members of the set A or “belong” to A.
Notation: ab A a A
3. V = {a, i, o, u, e} Set of Vowels
O = {1,3,5,7,9} Odd numbers < 10.
4. A1 = {2, 5, 8, 11, …, 101}
A2 = {1, 2, 3, 5, 8, 13,…}
A3 = {2, 5, 10, 17, 26, …, 101}
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1. Set Builder: B = {x | P(x)}
B1 = {x | x = n2 + 1, 1 n 10} (B1 = A3)
B2 = {p | p prime, p = n! + 1, n Special sets:
N (non-negative integers, natural numbers)
Q (rational numbers) Z (integers) Z+ (positive integers)
R (real numbers) (the empty set)
Basic notation.
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Relations among sets
1. A = B: if x A x B
2. Subsets: A B A B A B A B
3. For every set B: B
4. A set may have other sets as members:
A = {, {a}, {b}, {a,b}}. Note: A has 4 elements. A and also A, {{a}} A, a A, {a} A.
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Set operationsUnion: A B = {x | x A x B} (logic “or”)
Intersection: A B = {x | x A x B} ( “and”)
Set difference: A \ B = {x | x A x B}
Complement of A: A = {a | a A} or if U is the “universe” then A = U \ A (“not”).
Example: If U = {a,b,c,…,z} and A = {i,o,e,u,a} then A = {n | n is not a vowel}.
Symmetric difference: A B = (A \ B) (B \ A)(“xor”)
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The characteristic vector of a set (representing sets in memory):
Let U = {1,2,…, 15}. Let A = { 3,5,11,13} the characteristic vector of A is the binary string 00101 00000 10100.
The characteristic vector 10010 01101 10001 represents the set {1,4,7,8,10,11,15}.
00000 00000 00000 represents .
Note: with this representation the union of two sets is the OR bit operation and the intersection is the AND.
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A simple application.
Problem: find the smallest integer n that satisfies the following 3 conditions simultaneously:
(n mod 7 = 5), (n mod 11 = 7), (n mod 17 = 9)
Knowing the language “Math” can help us look for information and use various systems to solve this problem. The following exolains how to use SAGE's set operations to solve problems.
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We can create three sets:
1. A = {k | k = 7n + 5, k < 4000}
2. B = {k | k = 11n + 7, k < 4000}
3. C = {k | k = 17n + 9, k < 4000}
We can then ask SAGE to find the intersection of the three sets. The smallest integer in the intersection (provided there is one) will be our solution.
Answer: {502, 1811, 3120, ...}
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Venn Diagrams
Venn Diagrams : a useful tool for representing information. For instance, the various sets that can be formed by the basic set operations can be viewed by a Venn Diagram.
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A B
C
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Proving set equalities:
Either: x A x B or A B B A.
Example: De Morgan’s law: A B = A B
Proof: Let x A B.
Then: x A B.
Or: x A and x B
Or: x A and x B
Or: x A B
Conversely, start from the bottom and go up.
QED
Assume M = {1,2,5,9} then
= A1 A
2 A
5 A
9
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Notation:
A1 A
2 … A
n = {x | x A
i i = 1, 2, … , n}.
A1 A
2 … A
n = {x | ( i, 1i n) x A
i.
Use formula to insert intersection.
Assume M = {1,2,5,9} then
= A1 A
2 A
5 A
9
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The Power Set
Definition: The Power set of the set A is:
P(A) = {B | B A}.
has 0 elements. P() has one element: P() = {}
A = {a} P(A) = {, {a}} P({}) = {, {}}
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The cartesian product
Cartesian product :
A x B = {(a,b) | a A b B}
Can be defined using sets only:
A x B = {{a}, {a,b}| a A b B}
Note: (a,b) (b,a) if a b.
Cartesian product of n sets: A1x A
2 x … x A
n =
{(a1, a
2,…, a
n) | a
i A
i, i = 1,…,n}
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Relations
Definition 1: A relation R, (binary relation) between two sets A and B is a subset of
A x B (mathematically speaking: R A x B).
Definition 2: A relation R on a set A is a subset of A x A.
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Relations
There are two common ways to describe relations on a set or between two sets:
List all pairs belonging to the relation.
Use set builders to describe the pairs.
Example 1: R0 = {(4,3), (9,2), (3,6), (7,5)} is a
relation on N. It is also a relation on A x B where A = {4,9,3,7} and B = {3,2,6,5}
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More examples
Example 2: R2 = {(n,k) | n N and n + k is a
prime number}.
Example 3: R3 = {(n,k) | n,k N and |n – k| is a
multiple of 19}.
Example 4: R3 = {(w,m) | w is a woman, m is a
man, w dates m}
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Classification of relationsThese definitions apply to relations on A.
Definition 3: A relation R on A is reflexive
if (a,a) R a A.
Definition 4: A relation R on A is symmetric
if (a,b) R then (b,a) R.
R is antisymmetric
if (a,b) R and (b,a) R only if a = b.
Definition 5: A relation R on a set A is transitive
if (a,b) R (b,c) R then (a,c) R.
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The transitive closure
Observation: If R1 and R
2 are transitive
relations on a set A then so is R1 R
2.
Proof: Obvious.
Definition 6: The transitive closure of a relation R on a set A is the “smallest” transitive relation R* on A such that R* R.
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I think I solved it!