Sets and Propositions
Notion of a set set M
– collecti collection on of wel well-defi l-defined, ned, diffe differen rentt objects
elements
– obje objects of a set a ∈ M ⇐⇒ a belongs to the set M a∈ / M ⇐⇒ a does not belong to the set M
descri des cripti ption on – 1. by by enumer enumerati ation on of the elem elemen ents: ts: M = { a,b,c,...} 2. by characterizing the properties of elements with the help of a sentence form: M = { x ∈ Ω | A(x)true} empty emp ty set
– the set whi which ch does not con contai tain n any any elem elemen ent; t; notat notation ion:: ∅
disjoint disjo int sets sets – sets without without common common element elements: s: M ∩ N = ∅ Relations between sets Set inclusion (subset) M ⊂ N ⇐⇒ (∀ x : x ∈ M =⇒ x ∈ N N )) – M M subset subset of N N (inclusion) (inclusion) M ⊂ N ∧ (∃ x ∈ N : x : x ∈ / M M ))
– M M proper proper subset of N N
P (M M )) = { X | X ⊂ M }
– pow power se set, t, se sett of al alll su subs bset etss of the set M
Properties: M ⊂ M
– re refle flexi xivi vitty
M ⊂ N ∧ N ⊂ P =⇒ M ⊂ P
– tr tran ansi siti tivi vitty
∅ ⊂ M ∀ M
– ∅ is a subset of any set
N (proper subset: M ⊂ N N ). ). • Other notation of a subset: M ⊆ N Equality of sets M = N ⇐⇒ (∀ x : x ∈ M ⇐⇒ x ∈ N N )) – equality Properties: M ⊂ N ∧ N ⊂ M ⇐⇒ M = N
– or orde derr pr prope opert rty y
M = M
– re refle flexi xivi vitty
M = N =⇒ N = M
– sy sym mmet etry ry
M = = N ∧ N = = P =⇒ M = P
– tra trans nsit itiv ivit ity y
4
Sets and Propositions
Operations with sets M ∩ N = { x | x ∈ M ∧ x ∈ N } – intersection of the sets M and N ; contains all elements belonging both to M and to N (1) M ∪ N = { x | x ∈ M ∨ x ∈ N } – union of the sets M and N ; contains all elements belonging either to M or to N (or to both of them) (2) M \ N = { x | x ∈ M ∧ x ∈ / N } – difference of the sets M und N ; contains all elements of M not belonging to N (3) CΩ M = M = Ω \ M – complement to M with respect to Ω ; contains all elements of Ω not belonging to M , where Ω is some given basic set and M ⊂ Ω (4)
M
N
M
(1)
N
(2)
M
Ω
N
(3)
(4)
M CΩ M
• Sets M , N for which M ∩ N = ∅ (M, N having no elements in common) are called disjoint . • Operations with sets are also called connections between sets. Multiple connections
n
M i = M 1 ∪ M 2 ∪ . . . ∪ M n = { x | ∃ i ∈ {1, . . . , n } : x ∈ M i }
i=1 n
M i = M 1 ∩ M 2 ∩ . . . ∩ M n = { x | ∀ i ∈ {1, . . . , n } : x ∈ M i }
i=1
De Morgan’s laws M ∪ N = M ∩ N ,
M ∩ N = M ∪ N
(two sets)
n
n
(n sets)
i=1
M i =
n i=1
M i ,
i=1
M i =
n i=1
M i
Rules for operations with sets
5
Rules for operations with sets Union and intersection M ∪ (N ∩ M ) = M
M ∩ (N ∪ M ) = M
M ∪ (N ∪ P ) = (M ∪ N ) ∪ P
M ∩ (N ∩ P ) = (M ∩ N ) ∩ P
M ∪ (N ∩ P ) = (M ∪ N ) ∩ (M ∪ P ) M ∩ (N ∪ P ) = (M ∩ N ) ∪ (M ∩ P ) Union, intersection and difference M \ (M \ N ) = M ∩ N M \ (N ∪ P ) = (M \ N ) ∩ (M \ P ) M \ (N ∩ P ) = (M \ N ) ∪ (M \ P ) (M ∪ N ) \ P = (M \ P ) ∪ (N \ P ) (M ∩ N ) \ P = (M \ P ) ∩ (N \ P ) M ∩ N = ∅
⇐⇒
M \ N = M
Union, intersection and difference in connection with inclusion M ⊂ N ⇐⇒ M ∩ N = M M ⊂ N
=⇒
M ∪ P ⊂ N ∪ P
M ⊂ N
=⇒
M ∩ P ⊂ N ∩ P
⇐⇒ M ∪ N = N
M ⊂ N ⇐⇒ M \ N = ∅ Union, intersection and complement If both M ⊂ Ω and N ⊂ Ω , then the following relations hold (all complements taken with respect to Ω ):
∅ = Ω
Ω = ∅
M ∪ M = Ω
M ∩ M = ∅
M ∪ N = M ∩ N
M ∩ N = M ∪ N
(M ) = M
M ⊂ N ⇐⇒ N ⊂ M
De Morgan’s laws, s. p. 4
6
Sets and Propositions
Product sets and mappings Product sets ordered pair; combination of the elements x ∈ X , y ∈ Y in consideration of their order (x, y) = (z, w) ⇐⇒ x = z ∧ y = w – equality of two ordered pairs X × Y = { (x, y) | x ∈ X ∧ y ∈ Y } – product set, Cartesian product, cross or direct product (x, y)
–
Cross product of n sets n
X i = X 1 × X 2 × . . . × X n = { (x1 , . . . , x n ) | ∀ i ∈ {1, . . . , n} : xi ∈ X i }
i=1
X × X × . . . × X = X n ; n
times
IR × IR × . . . × IR = IRn
n
times
• The elements of X 1 × . . . × X n , i. e. (x1 , . . . , xn ), are called n-tuples , for n = 2 pairs , for n = 3 triples ; especially IR2 denotes all pairs, IRn all n-tuples of real numbers (vectors with n components). Mappings (relations) A ⊂ X × Y
–
mapping from X to Y ; subset of the cross product of the sets X and Y
DA = { x ∈ X | ∃ y : (x, y) ∈ A } –
domain of A
W A = { y ∈ Y | ∃ x: (x, y) ∈ A} –
range of A
A−1 = { (y, x) | (x, y) ∈ A}
reciprocal mapping; mapping inverse to the mapping A
–
• Let (x, y) ∈ A. Then y is an element associated with the element x. A mapping A from X to Y is called single-valued if for any element x ∈ X there is only one element y ∈ Y associated with x. A single-valued mapping is called a function f . The mapping rule is denoted by y = f (x). If both the mapping A and the inverse mapping A−1 (inverse function f −1 ) are single-valued, then A (and f , resp.) are called one-to-one mapping (function). Linear mapping f (λx + µy) = λf (x) + µf (y) – defining property of a linear mapping (function), λ, µ ∈ IR
• The composition h(x) = g(f (x)) of two linear mappings (e. g. f : IRn → IRm and g : IRm → IR p ) is again a linear mapping (h : IRn → IR p ) denoted by h = g ◦ f .
8
Sets and Propositions
• The implication (“from p it follows q ”) is also denoted as proposition in “if..., then...” form, p is called the premise (assumption), q is the conclusion (assertion). • The premise p is sufficient for the conclusion q , q is necessary for p. Other formulations for the equivalence are: “then and only then if . . . ” or “if and only if. . . (iff)”. Tautologies of propositional calculus p ∨ ¬ p
– law of excluded middle (excluded third)
¬ ( p ∧ ¬ p)
– law of contradiction
¬ (¬ p) ⇐⇒ p
– negation of the negation
¬ ( p =⇒ q ) ⇐⇒ ( p ∧ ¬ q )
– negation of the implication
¬ ( p ∧ q ) ⇐⇒ ¬ p ∨ ¬ q
– De Morgan’s law
¬ ( p ∨ q ) ⇐⇒ ¬ p ∧ ¬ q
– De Morgan’s law
( p =⇒ q ) ⇐⇒ ( ¬ q =⇒ ¬ p)
– law of contraposition
[( p =⇒ q ) ∧ (q =⇒ r)] =⇒ ( p =⇒ r)
– law of transitivity
p ∧ ( p =⇒ q ) =⇒ q
– rule of detachment
q ∧ (¬ p =⇒ ¬ q ) =⇒ p
– principle of indirect proof
[( p1 ∨ p2 ) ∧ ( p1 =⇒ q ) ∧ ( p2 =⇒ q )] =⇒ q – distinction of cases
Method of complete induction Problem: A proposition A(n) depending on a natural number n has to be proved for any n. Basis of the induction: The validity of the proposition A(n) is shown for some initial value (usually n = 0 or n = 1). Induction hypothesis: It is assumed that A(n) is true for n = k. Induction step: Using the induction hypothesis, the validity of A(n) is proved for n = k + 1.
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