Finite State Models

The University of Nottingham

School of Computer Science and IT

LECTURE 3 mathematical PRELIMINARIES PART 1 Overview
Sets, Relations and Functions Strings and Languages

Sets, Relations and Functions
A primed(p) is a collection of objects with no repetition. The simplest way to chance upon a set is by listing its elements. If a set is described using a defining property, the description should intelligibly specify the objects and the universe of discourse: A = {x | x ? ? ? x < 10} Important notation for sets: ?, a?A, a?A A?B, A?B, A?B, A=B, A?B, A?B, AB, A×B ?(A) or 2A, A or Ac , ?A what does this notation mean?

what do these notations mean?

For all sets A, B, and C in the universe U the following set properties hold: Associative uprightness: (A ? B) ? C = A ? (B ? C) (A ? B) ? C = A ? (B ? C) Commutative law: A?B=B?A A?B=B?A Complement law: A ? Ac = U A ? Ac = ?

1

Dr.



Dario Landa-Silva

The University of Nottingham

School of Computer Science and IT

Idempotency law:

A?A=A A?A=A

identity element law:

A??=A A?U=A

slide fastener law:

A?U=U A??=?

engagement law: De Morgans law:

(Ac)c = A (A ? B)c = Ac ? Bc (A ? B)c = Ac ? Bc

Distributive law:

A ? (B ? C) = (A ? B) ? (A ? C) A ? (B ? C) = (A ? B) ? (A ? C)

A set |A| is said to be finite if A contains a finite cast of elements. A set |A| is said to be infinite if A contains an infinite number of elements. The set A is said to be denumerable or enumerable is there is a way to list of the elements of A. more than formally, A set A is enumerable or countable is A is finite or if there is a bijection f:A??+. Example. The following sets are countable: explain wherefore?

A = {x | y ? ?+ ? x = 2?y+1} = {3, 5, 7, 9, 11, 13, 15,…} B = {(x,y) | x,y ? A} = {(3,3),(3,5),(5,3),(3,7),(5,5),(7,3),(3,9),(5,7),(7,5)…}

A relation R is a subset of A × B where A is the domain and B is the range of R. If the domain is the same as the range, A = B, so R is a relation on the set A. Properties of transaction:

2

Dr....If you want to get a full essay, order it on our website: Orderessay



If you want to get a full essay, wisit our page: write my essay .