Chapter 3 – Set Theory. 3.1 Sets and Subsets A set is a well-defined collection of objects. These...
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Transcript of Chapter 3 – Set Theory. 3.1 Sets and Subsets A set is a well-defined collection of objects. These...
3.1 Sets and Subsets A set is a well-defined collection of objects. These
objects are called elements and are said to be members of the set.
For a set A, we write x A if x is an element of A; y A indicated that y is not a member of A.
A set can be designated by listing its elements within set braces, e.g., A = {1, 2, 3, 4, 5}.
Another standard notation for this set provides us with A = {x | x is an integer and 1 x 5}. Here the vertical line | within the set braces is read “such that”. The symbols {x |…} are read “the set of all x such …”. The properties following | help us determine the elements of the set that is being described.
Example 3.2 page 128 From the above example, A and B are
examples of finite sets, where C is an infinite set. For any finite set A, |A| denotes the number of elements in A and is referred to as the cardinality, or size, of A, e.g., |A| = 9, |B| = 4.
Definition 3.1: If C, D are sets from a universe U, we say that C is a
subset of D and write C D, or D C, if every element of C is an element of D. If, in addition, D contains an element that is not in C, then C is called a proper subset of D, and this is denoted by C D or D C.
Note: 1) For all sets C, D from a universe U, if C D, then x [x C x D],
and if x [x C x D], then C D. That is, C D x [x C x D]. 2) For all subsets C, D of U, C D C D. 3) When C, D are finite, CD |C||D|, and
CD |C|<|D|.
Definition 3.2: For a given universe U, the sets C and
D (taken from U) are said to be equal, and we write C = D, when C D and D C.
Note: Some notions from logic: page 130 (line 4 from top).
Theorem 3.1: Let A, B, C U, a) If AB and BC, then AC. b) If AB and BC, then AC. c) If AB and BC, then AC. d) If AB and BC, then AC.
Definition 3.3: The null set, or empty set, is the
(unique) set containing no elements. It is denoted by or { }. (Note that ||=0 but {0}. Also, {} because {} is a set with one element, namely, the null set.)
Definition 3.4: If A is a set from universe U, the power
set of A, denoted (A), is the collection (or set) of all subsets of A.
Lemma: For any finite set A with |A| = n 0,
we find that A was 2n subsets and that |(A)| = 2n. For any 0 k n, there are subsets of size k. Counting the subsets of A according to the number, k, of elements in a subset, we have the combinatorial identity , for n 0.
nn
k k
n
n
nnnn2...
210 0
Definition 3.5: For A, B U we define the followings: A B (the union of A and B) = {x | x A x B }. A B (the intersection of A and B) = {x | x A x
B }. A B (the symmetric difference of A and B) = {x | (x
A xB) xAB} = {x | xAB xAB}. Note: If A, B U, then A B, A B, A B U. C
onsequently, , , and are closed binary operations on (A), and we may also say that (A) is closed under these (binary) operations.
Definition 3.6: Let S, T U. The sets S and T are
called disjoint, or mutually disjoint, when S T = .
Theorem 3.3: If S, T U, then S and T are disjoint if and
only if S T = S T. proof) proof by contradiction.
Definition 3.8: A, B U, the (relative) complement of
A in B, denoted B – A, is given by {x | xB xA}.
Theorem 3.4: For any universe U and any sets A, B
U, the following statements are equivalent:
a) A B b) A B = B c) A B = A d) B’ A’
Definition 3.9: Let s be a (general) statement dealing with the
equality of two set expressions. Each such expression may involve one or more occurrences of sets (such as A, , B, , etc.), one or more occurrences of and U, and only the set operation symbols and . The dual of s, denoted sd, is obtained from s by replacing (1) each occurrence of and U (in s) by U and , respectively; and (2) each occurrence of and (in s) by and , respectively.
Theorem 3.5: The Principle of Duality.
Let s denote a theorem dealing with the equality of two set expressions (involving only the set operations and as described in Definition 3.9). Then sd, the dual of s, is also a theorem.
Venn diagram Venn diagram is constructed as
follows: U is depicted as the interior of a rectangle, while subsets of U are represented by the interiors of circles and other closed curves. (See Fig 3.5 and 3.6, page 145.)
Membership table: We observe that for sets A, B U, an elem
ent xU satisfies exactly one of the following four situations:
a) xA, xB b) xA, xB c) xA, xB d) xA, xB. When x is an element of a given set, we writ
e a 1 in the column representing that set in the membership table; when x is not in the set, we enter a 0. See Table 3.2 and 3.3, page 147.
(1) A Venn diagram is simply a graphical representation of a membership table.
(2) The use of Venn diagrams and/or membership tables may be appealing, especially to the reader who presently does not appreciate writing proofs.
Fig 3.8 (page 152) demonstrates and , so by the rule of sum, |A| + || = |U| or || = |U| - |A|. If the sets A, B have empty intersection, Fig 3.9 shows |A B| = |A| + |B|; otherwise, |A B| = |A| + |B| - | A B| (Fig 3.10).
Lemma: If A and B are finite sets, then |A B| = |A| + |B| - | A B|. Consequently, finite sets A and B are (m
utually) disjoint if and only if |A B| = |A| + |B|.
In addition, when U is finite, from DeMorgan’s Law we have
|| = || = |U| - |A B| = |U| - |A| - |B|+|A B|.
Lemma: If A, B, C are finite sets, then . From the formula for |A B C| and
DeMorgan’s Law, we find that if the universe U is finite, then
Example 3.25: page 153~154.
Lemma: Under the assumption of equal likelihoo
d, let Φ be a sample space for an experiment Ε. Any subset A of Φ is called an event. Each element of Φ is called an elementary event, so if |Φ| = n and a Φ, A Φ, then
Pr(a) = The probability that a occurs =, and
Pr(A) = The probability that A occurs =.