Two sets are said to be disjoint if their intersection is empty; that is, if they have no
elements in common. A collection {Ai : i 2 I} of sets is said to be disjointed if Ai
and Aj are disjoint for all i and j in I with i 6= j.
Products of Sets
Let A and B be sets. Their product, denoted by A×B, is the set of all pairs (x, y) with
x in A and y in B. It is also called the rectangle with sides A and B.
If A1, . . . ,An are sets, then their product A1 × · · · × An is the set of all n-tuples
(x1, . . . , xn) where x1 2 A1, . . . , xn 2 An. This product is called, variously, a rectangle,
or a box, or an n-dimensional box. If A1 = · · · = An = A, then A1 × · · · × An
is denoted by An. Thus, R2 is the plane, R3 is the three-dimensional space, R2
+ is the
positive quadrant of the plane, etc.
Exercises:
1.1 Let E be a set. Show the following for subsets A,B,C, and Ai of E.
Here, all complements are with respect to E; for instance, Ac = E \ A.
1. (Ac)c = A
2. B \ A = B \ Ac
3. (B \ A) \ C = (B \ C) \ (A \ C)
4. (A [ B)c = Ac \ Bc
5. (A \ B)c = Ac [ Bc
6. (
S
i2I Ai)c =
T
i2I Aci
7. (
T
i2I Ai)c =
S
i2I Aci
1.2 Let a and b be real numbers with a < b. Find
1[
n=1
[a +
1
n
, b ?
1
n
],
1\
n=1
[a ?
1
n
, b +
1
n
]
4 SETS AND FUNCTIONS
1.3 Describe the following sets in words and pictures:
1. A = {x 2 R2 : x21
+ x22
< 1}
2. B = {x 2 R2 : x21
+ x22
1}
3. C = B \ A
4. D = C × B
5. S = C × C
1.4 Let An be the set of points (x, y) 2 R2 lying on the curve y = 1/xn,
0 < x < 1. What is
T
n1 An?
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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