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. Object a is not an element of a set A: a / ∈ A. Important sets:

• natural numbers N = {1, 2, 3, . . . }, also N0 = {0, 1, 2, 3, . . . };
• integer numbers (integers) Z = {0, 1, −1, 2, −2, 3, . . . };
• rational numbers Q =

p

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.

p q p ∧ q 1 1 1 1 0 0 1 0 0 p q p ∨ q 1 1 1 1 0 1 0 1 1 0 0 p q p = ⇒ q 1 1 1 1 0 0 1 1 0 0 1 p q p ⇐ ⇒ q 1 1 1 1 0 0 1 0 0 1

Definition. Let A be a subset in some universe U. We define its complement with respect to U as Ac = A = {x ∈ U; x / ∈ A}. Definition. Let A, B be sets in some universe U. We define their union: A ∪ B = {x ∈ U; x ∈ A ∨ x ∈ B}; intersection: A ∩ B = {x ∈ U; x ∈ A ∧ x ∈ B}; difference or complement of B in A: A − B = {x ∈ U; x ∈ A ∧ x / ∈ B}; Cartesian product: A × B = {(a, b); a ∈ A ∧ b ∈ B}, 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 Ai for i ∈ I be sets in the same universe U, where I is some set

• f indices. We define
• i∈I

Ai = {x ∈ U; ∃i ∈ I : x ∈ Ai},

• i∈I

Ai = {x ∈ U; ∀i ∈ I : x ∈ Ai}.

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

• nly 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

• f 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 N0 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 An for n ∈ N are at most countable, then

∞

• n=1

An is also at most countable. (ii) If moreover the sets An are non-empty and pairwise disjoint, then

∞

• n=1

An is countable.

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)|.