Download This document contains information about theory if automata and more Schemes and Mind Maps Theory of Automata in PDF only on Docsity! SET Subsets • The empty set is a subset of every set • Every set is a subset of itself • The power set of a set A denoted by 2A is the set contains all possible subsets of set A • Example A = {1,2,3} 2A = { { }, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3} } • = Properties of Set Let A, B, C be sets in the universal set U • Idempotency: A U A = A A ∩ A = A • Identity: A U φ = A A ∩ U = A • Zero: A U U = U A ∩ φ = φ • Involution: (A )′ ′ = A Properties of Set Let A, B, C be sets in the universal set U • DeMorgan’s: (A U B)′ = A ∩ B ′ ′ (A ∩ B) = A U B ′ ′ ′ • Distributivity: A U (B ∩ C) = (A U B) ∩ (A U C) • True for all operators • Zero: A U U = U A ∩ φ = φ • Involution: (A ) = A′ ′ Relations • A Binary relation R on two sets A and B is a subset of A X B Example: Let A = {1, 4}, B = {1,5} A x B? • R1 is such that 1st number < 2nd number • R2 is such that the two numbers are equal Properties of Relations • Symmetric: A relation R is symmetric if for any aRb in R, there is bRa in R • Let L be the relation “ < “ • Let E be the relation “ = “ • Let P be the relation “ the two numbers are either both even or both odd“ : Reflexive Symmetric L X X E ✓ ✓ P ✓ ✓ Properties of Relations • Transitive: A relation R is Transitive if we have aRb and bRc, and there is also aRc. • Let L be the relation “ < “ • Let E be the relation “ = “ • Let P be the relation “ the two numbers are either both even or both odd“ : Reflexive Symmetric Transitive L X X ✓ E ✓ ✓ ✓ P ✓ ✓ ✓ Properties of Relations • Antisymmetric: if we have aRb and bRa, then a = b a,b Ɐ ϵ A. • Let L be the relation “ < “ • Let E be the relation “ = “ • Let P be the relation “ the two numbers are either both even or both odd“ : Reflexive Symmetric Transitive Antisymmetric L X X ✓ ✓ E ✓ ✓ ✓ ✓ P ✓ ✓ ✓ X Functions: Total vs Partial • Total function: If f(x) is defined for all x in A, then f is a total function • Partial function: If f(x) is not defined for some elements in A, then f is a partial function Functions: Onto vs One to one • Onto: f(x) is onto if all elements of B are images • One to one: f(x) is one to one if unique elements in A maps to unique elements in B. • Not a function: if any element in A maps to more than one elements in B • Bijection: a function that is total, onto, and one to one
