Chapter 6 Set Theory

Axioms of Zermelo-Fraenkel 1. Axiom of Extensionality 2. Axiom of Pairing 3. Axiom Schema of Separation. If is a property with parameter p, then for any set X, there exists a set Y = { u X: (u, p)} 4. Axiom of Union For any set X, there is a set Y that is the union of all elements in X. 5. Axiom of Power Set For any set X, there exists a set Y that contains all subsets of X. 6. Axiom of Infinity There is an infinite set. 7. Axiom Schema of Replacement If F is a function, then for any set X, there is a set Y = {F(x): x X} 8. Axiom of Regularity Every non-empty set has an -minimal element. 9. Axiom of Choice For every non-empty set S of non-empty sets, there is at least one choice function for S.
ExeБайду номын сангаасcises
Determine whether each statement is true or false.
a) 3 {1,2,3} b) 1 {1} c) {2} {1, 2} True False False f) {2} {1, {2},{3}} g) {1} {1, 2} h) 1 {{1}, 2} False
A B {( a, b) : a A and b B}
Given sets A1, A2 , … , An we can define the Cartesian product
A1 × A2 × · · An ·×
as the set of all ordered n-tuples.
6.1 Basic Definitions
Cartesian Products: For any two sets A and B, the Cartesian Product of A and B, denoted by A×B (read A cross B), is the set of all ordered pairs of the form (a, b) where aA and bB.
True False
d) {3} {1, {2}, {3}}
e) 1 {1} True
True
i) {1} {1, {2}}
j) {1} {1}
True
True
6.1 Basic Definitions
Operations of Sets: Let A and B be two subsets of a larger set U, we can define the following, 1. Union of A and B,
6.1 Basic Definitions
Introduction:
The { } notation: If a set M has only a finite number of elements say, 3, 7, and 11, then we can write M = {3, 7, 11} (the order in which they appear is unimportant.) A set can also be specified by a defining property, for instance S = {x : -2 < x < 5} (this is almost always the way to define an infinite set).
A B {x U : x A or x B}
2. Intersection of A and B,
A B {x U : x A and x B}
3. Difference of B minus A,
B A {x U : x B and x A}
4. Complement of A,
Chapter 6 Set Theory
6.1 Basic Definitions
Introduction:
Roughly speaking, a set is a collection of objects that satisfy a certain property, but in set theory, the words “set” and “element” are intentionally left as undefined. There is also another undefined relation , called the “membership” relation. If S is a set and a is an element of S, then we write a S, and we can say that a belongs to S.
6.1 Basic Definitions
Subsets: Given two sets A and B, we say that A is a subset of B, denoted by
A B
if x ( x A x B ) In other words, A is a subset of B if all elements in A are also in B.
A B (A B ) (B A)
Proper Subsets: Given two sets A and B, we say that A is a proper subset of B, denoted by
A B
if
A B and A B
In other words, A is a proper subset of B if all elements in A are also in B but A is “smaller” than B.
AC {x U : x A}
6.1 Basic Definitions
Ordered pairs: For any two elements a and b (not necessarily distinct), we define (a, b) = {a, {a, b}} which is called the ordered pair of a and b. It is not hard to see that (a, b) = (c, d) if and only if (a = c and b = d) In other words, the order of the elements is important. We can similarly define ordered triples and ordered n-tuples (x1, x2, x3, … , xn)
H = { a, e, i, o, u, h, k, l, m, n, p, w}
Hence H is a subset of E.
6.1 Basic Definitions
Lemma: Two sets A and B are equal if A is a subset of B and B is a subset of A. i.e.
Definition of whole numbers
The number 0 is defined as the empty set Ø. The number 1 is defined as the set {Ø}. The number 2 is defined as the set {Ø} {{Ø}} = {Ø, {Ø}} … … In general, the natural number n + 1 is defined as the set n {n} We can prove by induction that the set representing n has exactly n elements.
Example: (see next page)
An example of subset: Let E be the English alphabet, hence it is a set of 26 letters E = {a, b, c, … , x, y , z}
In the Hawaiian alphabet however, it contains only 7 consonants
A B x ( x A x B)
In other words, two sets are equal if and only if they have the same elements.
For example, {1, 2, 3} = {3, 1, 2} = {1, 2, 3, 2}
Choice Function
Let S be a collection of non-empty sets, then a choice function for S is a function
f :S X
X S