المحاضرة الاولى- احتسابية1

Computation theory
Computations are designed to solve problems. Computations are designed for processing information. They can be as simple as estimation for driving time between cities, and as complex as a weather prediction.
In theoretical computer science and mathematics, the theory of computation is the branch that deals with how efficiently problems can be solved on a model of computation, using an algorithm. The field is divided into three major branches: automata theory and languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers?"
The theory of computation can be considered the creation of models of all kinds in the field of computer science. Therefore, mathematics and logic are used. In the last century it became an independent academic discipline and was separated from mathematics.

1-Sets
A set is a collection of "things" called the elements or members of the set.
Common forms of describing sets are:
• List all the elements, e.g. {a, b, c, d}
• Form new sets by combining sets through operators (see next page).
Terminology and Notation:
• To indicate that x is a member of set S, we write x?S.
• We denote the empty set (the set with no members) as { } or ?.
• If every element of set A is also an element of set B, we say that A is a subset of B, and write A? B
• If every element of set A is also an element of set B, but B also has some elements not contained in A, we say that A is a proper subset of B, and write A?B
• We may also use the inverse notation: B?A and B?A for B is a (proper) superset of A.

Note: It is essential to have a criterion for determining, for any appropriate “thing”, whether it is or is not a member of the given set. This is called the membership criterion. Languages – which we will introduce later - are sets. These sets contain specific strings over an alphabet, according to certain specifications or conditions, which describe the language. Grammars and automata can be used to describe languages (and therefore also sets). Membership criteria and appropriate decision algorithms are a central topic in the study of formal languages.

Operations on Sets
???The union of sets A and B, written
Computation theory
Computations are designed to solve problems. Computations are designed for processing information. They can be as simple as estimation for driving time between cities, and as complex as a weather prediction.
In theoretical computer science and mathematics, the theory of computation is the branch that deals with how efficiently problems can be solved on a model of computation, using an algorithm. The field is divided into three major branches: automata theory and languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers?"
The theory of computation can be considered the creation of models of all kinds in the field of computer science. Therefore, mathematics and logic are used. In the last century it became an independent academic discipline and was separated from mathematics.

1-Sets
A set is a collection of "things" called the elements or members of the set.
Common forms of describing sets are:
• List all the elements, e.g. {a, b, c, d}
• Form new sets by combining sets through operators (see next page).
Terminology and Notation:
• To indicate that x is a member of set S, we write x?S.
• We denote the empty set (the set with no members) as { } or ?.
• If every element of set A is also an element of set B, we say that A is a subset of B, and write A? B
• If every element of set A is also an element of set B, but B also has some elements not contained in A, we say that A is a proper subset of B, and write A?B
• We may also use the inverse notation: B?A and B?A for B is a (proper) superset of A.

Note: It is essential to have a criterion for determining, for any appropriate “thing”, whether it is or is not a member of the given set. This is called the membership criterion. Languages – which we will introduce later - are sets. These sets contain specific strings over an alphabet, according to certain specifications or conditions, which describe the language. Grammars and automata can be used to describe languages (and therefore also sets). Membership criteria and appropriate decision algorithms are a central topic in the study of formal languages.

Operations on Sets
???The union of sets A and B, written
Computation theory
Computations are designed to solve problems. Computations are designed for processing information. They can be as simple as estimation for driving time between cities, and as complex as a weather prediction.
In theoretical computer science and mathematics, the theory of computation is the branch that deals with how efficiently problems can be solved on a model of computation, using an algorithm. The field is divided into three major branches: automata theory and languages, computability theory, and computational complexity theory, which are linked by the question: "What are the fundamental capabilities and limitations of computers?"
The theory of computation can be considered the creation of models of all kinds in the field of computer science. Therefore, mathematics and logic are used. In the last century it became an independent academic discipline and was separated from mathematics.

1-Sets
A set is a collection of "things" called the elements or members of the set.
Common forms of describing sets are:
• List all the elements, e.g. {a, b, c, d}
• Form new sets by combining sets through operators (see next page).
Terminology and Notation:
• To indicate that x is a member of set S, we write x?S.
• We denote the empty set (the set with no members) as { } or ?.
• If every element of set A is also an element of set B, we say that A is a subset of B, and write A? B
• If every element of set A is also an element of set B, but B also has some elements not contained in A, we say that A is a proper subset of B, and write A?B
• We may also use the inverse notation: B?A and B?A for B is a (proper) superset of A.

Note: It is essential to have a criterion for determining, for any appropriate “thing”, whether it is or is not a member of the given set. This is called the membership criterion. Languages – which we will introduce later - are sets. These sets contain specific strings over an alphabet, according to certain specifications or conditions, which describe the language. Grammars and automata can be used to describe languages (and therefore also sets). Membership criteria and appropriate decision algorithms are a central topic in the study of formal languages.

Operations on Sets
???The union of sets A and B, written