A set A is said to be a subset of a set B if every element of A is an element of B. We
write A B or B A to indicate it and use expressions like A is contained in B,
B contains A, to the same effect. The sets A and B are the same, and then we write
A = B, if and only if A B and A B. We write A 6= B when A and B are not the
same. The set A is called a proper subset of B if A is a subset of B and A and B are
not the same.
The empty set is a subset of every set. This is a point of logic: let A be a set;
the claim is that ; A, that is, that every element of ; is also an element of A,
or equivalently, there is no element of ; that does not belong to A. But the last is
obviously true simply because ; has no elements.
Set Operations
Let A and B be sets. Their union, denoted by A[B, is the set consisting of all elements
that belong to either A or B (or both). Their intersection, denoted by A \ B, is the
set of all elements that belong to both A and B. The complement of A in B, denoted
by B \ A, is the set of all elements of B that are not in A. Sometimes, when B is
understood from context, B \ A is also called the complement of A and is denoted by
Ac. Regarding these operations, the following hold:
Commutative laws:
A [ B = B [ A,
A \ B = B \ A.
Associative laws:
(A [ B) [ C = A [ (B [ C),
(A \ B) \ C = A \ (B \ C).
Distributive laws:
A \ (B [ C) = (A \ B) [ (A \ C),
A [ (B \ C) = (A [ B) \ (A [ C).
The associative laws show that A[B[C and A\B\C have unambiguous meanings.
Definitions of unions and intersections can be extended to arbitrary collections of
sets. Let I be a set. For each i 2 I, let Ai be a set. The union of the sets Ai, i 2 I, is
the set A such that x 2 A if and only if x 2 Ai for some i in I. The following notations
are used to denote the union and intersection respectively:
[
i2I
Ai,
\
i2I
Ai.
1. SETS 3
When I = N = {1, 2, 3, . . .}, it is customary to write
المادة المعروضة اعلاه هي مدخل الى المحاضرة المرفوعة بواسطة استاذ(ة) المادة . وقد تبدو لك غير متكاملة . حيث يضع استاذ المادة في بعض الاحيان فقط الجزء الاول من المحاضرة من اجل الاطلاع على ما ستقوم بتحميله لاحقا . في نظام التعليم الالكتروني نوفر هذه الخدمة لكي نبقيك على اطلاع حول محتوى الملف الذي ستقوم بتحميله .
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