in. zagadnienia takie jak: iloczyn kartezjański, relacja zwrotna, relacja autosymetryczna, relacja przeciwsymetryczna, relacja przeciwzwrotna, relacja jednoznaczna, zbiór, ciąg, indukcja, symbol dwumianowy. W notatce znaleźć można równie...
An introduction to the graph theory A graph is a set of points, together with a set of arcs that connect pairs of these points. It can be more than one arc connecting the same pair of points. An arc can connect a point to itself, forming a loop. A graph can be described by giving the set: V = { v 1...
Binary relation Definition Let A and B be nonempty sets. A binary relation from A to B is a subset of the Cartesian Product A × B . A binary relation from A to A is called a binary relation on A . Let R ⊂ A × B ( R is a binary relation from A to B ). If ( a, b ) ∈ R , we say: a is related to b and ...
Boolean function and conjunctive normal form. A Boolean function p ( x 1 , x 2 , ..., x n ) is said to be in conjunctive normal form , if it is the conjunction of finite number of distinct terms, each of which has the form e 1 ∨ e 2 ∨ ... ∨ e n , where e i = x i or e i = x ′ i for all i = 1 , 2 , ,...
Boolean function and disjunctive normal form. Let x 1 , x 2 , ..., x n be a sequence of statement variables. Then any compound statement of these variables will be called a Boolean polynomial or Boolean function . p ( x 1 , x 2 , ..., x n ) - Boolean polynomial of variables x 1 , x 2 , ..., x n . 0...
Tautology and contradiction Theorem Let A and B be compound statements. If the compound statement A and A = ⇒ B are tautologies, then so is the compound statement B . The substitution theorem for tautologies Let A be a tautology and suppose that A contains the distinct statement variables p 1 , p 2...
Countable and uncountable sets. There is a one-to- one correspondence between sets N (all natural numbers) and N 2 (even natural numbers). The correspondence has the form f ( n ) = 2 n . There is a one-to-one correspondence between sets N and N odd (odd natural numbers). The correspondence has the ...
Elements of mathematical proving. If p = ⇒ q ( ⋆ ) is a statement (let's call it a simple statement ), then there are 3 other statements related to this. q = ⇒ p - converse of statement ( ⋆ ) ; ( ∼ p ) = ⇒ ( ∼ q ) - inverse of statement ( ⋆ ) ; ( ∼ q ) = ⇒ ( ∼ p ) - contrapositive of statement ( ⋆ ...
Extremal elements Let _ be a partial order on P ⊂ X . Element a 0 ∈ P is the largest element in P , if a 0 is greater than every other element in P : ∀ a ∈ P ( a 6 = a 0 ) a ≺ a 0 . Element a 1 ∈ P is the smallest element in P , if a 1 is less than every other element in P : ∀ a ∈ P ( a 6 = a 1 ) a...
Function as a relation A subset R of the cartesian product X × Y is called a binary relation defined on elements of sets X and Y . Definition Let X and Y be any sets. A relation f which satisfies the following two conditions: ∀ x ∈ X ∃ y ∈ Y ( x, y ) ∈ f ; ∀ x ∈ X ∀ y 1 , y 2 ∈ Y [( x, y 1 ) ∈ f ∧ ...
