By a set we mean any collection of objects that are precisely spec- - PDF document

By a set we mean any collection of objects that are precisely spec- ified. These objects are called members or elements of the given set. Every set is uniquely determined by these objects; that is, if two sets contain the same objects, then they are the same set. Notation: Object a is an element of a set A : a ∈ A . ∈ A . Object a is not an element of a set A : a / Important sets: • natural numbers N = { 1 , 2 , 3 , . . . } , also N 0 = { 0 , 1 , 2 , 3 , . . . } ; • integer numbers (integers) Z = { 0 , 1 , − 1 , 2 , − 2 , 3 , . . . } ; � p � • rational numbers Q = q ; p ∈ Z ∧ q ∈ N ; • real numbers R ; • empty set, a set without elements: ∅ = {} .

Definition. Let A, B be set. We say that A is a subset of B , denoted A ⊆ B , if ∀ a ∈ A : a ∈ B .

Fact. Let A be a set. (i) A ⊆ A ; (ii) ∅ ⊆ A . Fact. Let A, B, C be sets. If A ⊆ B and B ⊆ C , then A ⊆ C .

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Definition. Let A be a subset in some universe U . We define its complement with respect to U as A c = A = { x ∈ U ; x / ∈ A } . Definition. Let A, B be sets in some universe U . We define their A ∪ B = { x ∈ U ; x ∈ A ∨ x ∈ B } ; union: A ∩ B = { x ∈ U ; x ∈ A ∧ x ∈ B } ; intersection: difference or complement of B in A : A − B = { x ∈ U ; x ∈ A ∧ x / ∈ B } ; A × B = { ( a, b ); a ∈ A ∧ b ∈ B } , Cartesian product: here ( a, b ) denotes an ordered pair.

Theorem. (laws for set operations) Let A, B, C be arbitrary sets from a universe U . Then the following are true: (i) A ∪ ∅ = A , A ∩ U = A ; (identity laws) (ii) A ∩ ∅ = ∅ , A ∪ U = U ; (cancellation laws) (iii) A ∪ A = A , A ∩ A = A ; (idempotence laws) (iv) A = A ; (double complement law) (v) A ∪ B = B ∪ A , A ∩ B = B ∩ A ; (commutative laws) (vi) A ∪ ( B ∪ C ) = ( A ∪ B ) ∪ C , A ∩ ( B ∩ C ) = ( A ∩ B ) ∩ C ; (associative laws) (vii) A ∩ ( B ∪ C ) = ( A ∩ B ) ∪ ( A ∩ C ), A ∪ ( B ∩ C ) = ( A ∪ B ) ∩ ( A ∪ C ); (distributive laws) (viii) A ∪ B = A ∩ B , A ∩ B = A ∪ B ; (De Morgan’s laws) (ix) A ∪ ( A ∩ B ) = A , A ∩ ( A ∪ B ) = A ; (absorbtion laws) (x) A ∪ A = U , A ∩ A = ∅ . (complement laws)

• ¬ ( ¬ p ) | | p ; = • ¬ ( p ∧ q ) | = | ¬ p ∨ ¬ q ; • ¬ ( p ∨ q ) | = | ¬ p ∧ ¬ q ; • ¬ ( p = ⇒ q ) | = | p ∧ ¬ q ; • ¬ ( p ⇐ ⇒ q ) | | ( p ∧ ¬ q ) ∨ ( q ∧ ¬ p ). = • ¬ ( ∀ x ∈ M : p ( x )) | | ∃ x ∈ M : ¬ p ( x ); = • ¬ ( ∃ x ∈ M : p ( x )) | = | ∀ x ∈ M : ¬ p ( x ).

Definition. Let A i for i ∈ I be sets in the same universe U , where I is some set of indices. We define � A i = { x ∈ U ; ∃ i ∈ I : x ∈ A i } , i ∈ I � A i = { x ∈ U ; ∀ i ∈ I : x ∈ A i } . i ∈ I

Definition. Sets A, B are called disjoint if A ∩ B = ∅ .

Definition. Let A, B be non-empty sets. By a mapping from A to B we mean arbitrary subset of A × B that satisfies the condition ∀ a ∈ A ∃ ! b ∈ B : ( a, b ) ∈ T. The set A is called the domain of T , denoted D ( T ), the set B is the codomain of T . We also define the range of T as R ( T ) = { b ∈ B ; ∃ a ∈ A : T ( a ) = b } = { T ( a ); a ∈ A } .

Definition. Let T : A �→ B and S : C �→ D be mappings. We say that they are equal, denoted T = S , if A = C , B = D , and ∀ a ∈ A : T ( a ) = S ( a ) .

Definition. Let T : A �→ B and S : B �→ C be mappings. We define their composite mapping or composed mapping or composition S ◦ T : A �→ C by the formula ( S ◦ T )( a ) = S ( T ( a )) for a ∈ A. We also denote S ◦ T = S ( T ).

Theorem. Let T : A �→ B , S : B �→ C , and R : C �→ D be mappings. Then ( R ◦ S ) ◦ T = R ◦ ( S ◦ T ).

Definition. Let T : A �→ B be a mapping. We say that a mapping S : B �→ A is an inverse mapping of T if the following are true: • ( S ◦ T )( a ) = a for all a ∈ A , • ( T ◦ S )( b ) = b for all b ∈ B . If such a mapping exists, then we say that T is invertible and denote that inverse mapping as T − 1 .

Fact. Let T : A �→ B be an invertible mapping. Then T − 1 ( b ) = a if and only if T ( a ) = b .

Corollary. Let T : A �→ B be a mapping. If it is invertible, then the inverse mapping T − 1 is unique.

Theorem. Let T : A �→ B and S : B �→ C be mappings. If they are invertible, then also S ◦ T is invertible and ( S ◦ T ) − 1 = T − 1 ◦ S − 1 .

Definition. Let T : A �→ B be a mapping. We say that T is one-to-one or injective if ∀ x, y ∈ A : x � = y = ⇒ T ( x ) � = T ( y ) . We say that T is onto or surjective if R ( T ) = B . We say that T is bijective or a bijection if it is 1-1 and onto.

Definition. Let T : A �→ B be a mapping. We say that T is one-to-one or injective if ∀ x, y ∈ A : x � = y = ⇒ T ( x ) � = T ( y ) . We say that T is onto or surjective if R ( T ) = B . We say that T is bijective or a bijection if it is 1-1 and onto. Alternative definition of one-to-one: ∀ x, y ∈ A : T ( x ) = T ( y ) = ⇒ x = y.

Theorem. Let T : A �→ B be a mapping. It is invertible if and only if it is a bijection.

Fact. Consider mappings T : A �→ B and S : B �→ C . The following are true: (i) If T and S are 1-1, then also S ◦ T is 1-1. (ii) If T and S are onto, then also S ◦ T is onto. (iii) If T and S are bijective, then also S ◦ T is a bijection.

Fact. Let T : A �→ B be a mapping, assume that A, B have finitely many elements. (i) If B has more elements than A , then T can never be onto. (ii) If A has more elements than B , then T can never be 1-1. (iii) If A and B do not have the same number of elements, then T cannot be a bijection.

Definition. We say that sets A, B have the same cardinality , denoted | A | = | B | , if there exists a bijection from A to B . We say that the set A has cardinality greater or equal to cardinality of B , denoted | A | ≤ | B | , if there exists a 1-1 mapping from A to B .

Fact. Consider arbitrary sets A, B . The following are true: (i) | A | = | B | if and only if | B | = | A | . (ii) If | A | = | B | , then | A | ≤ | B | and | B | ≤ | A | .

Theorem. (Cantor-Bernstein-Schroeder) Let A, B be sets. If | A | ≤ | B | and | B | ≤ | A | , then | A | = | B | .

Definition. A set A is called finite if A = ∅ (then we write | A | = 0) or there exists m ∈ N such that | A | = |{ 1 , 2 , . . . , m }| , then we write | A | = m . Otherwise we call this set infinite . A set A is called countable if it has the same cardinality as the set N . A set A is called uncountable if it is infinite but not countable.

Fact. Let A be a set. If it is infinite, then | N | ≤ | A | .

Theorem. (i) If A is a finite set, then also every its subset B is finite and | B | ≤ | A | . Moreover, if B is a proper subset, then | B | < | A | . (ii) Let A, B be finite sets. Then A ∪ B is also finite and | A ∪ B | ≤ | A | + | B | . If moreover A, B are disjoint, then | A ∪ B | = | A | + | B | . (iii) Let A, B be finite sets. Then A × B is also finite and | A × B | = | A | · | B | .

Theorem. (i) Every infinite set has a proper subset which has the same cardi- nality. (ii) Let A, B be sets, assume that A is infinite and | B | ≤ | A | . Then | A ∪ B | = | A | . (iii) Let A, B be sets. Assume that A is infinite and | B | ≤ | A | . Then | A × B | = | A | .

Theorem. (i) The set N 0 is countable. (ii) The set Z is countable. (iii) The set N × N is countable. (iv) The set Z × Z is countable.

Theorem. The set Q of all rational numbers is countable.

Theorem. The interval � 0 , 1) of real numbers is uncountable.

Corollary. The set R of all real numbers is uncountable.

Fact. ∞ (i) If sets A n for n ∈ N are at most countable, then � A n is also at n =1 most countable. (ii) If moreover the sets A n are non-empty and pairwise disjoint, then ∞ � A n is countable. n =1

Definition. Let A be a set. We define the power set (or powerset ) of A , denoted P ( A ), as the set of all subsets of A .

Fact. If A is a finite set, then | P ( A ) | = 2 | A | .

Theorem. (Cantor) For every set A the following is true: | A | < | P ( A ) | .