By Peter W. And Johnson, Robert Zehna

A really brief creation to Set thought for undergraduate scholars.

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**Example text**

5 • 43 which may help to point out other generalizations and which will make many of the conclusions at least intuitively appealing. This device is a simple diagrammatical portrayal of a universe of discourse and some of its subsets. A universe of discourse is represented by a large rectangular area and subsets within this universe are represented by circular areas. If a certain set is only partially in the universe of discourse, it may be represented by a circular area part of which is in the rectangle.

In the following, decide whether the statement is true or false. If true, show why. , an instance when the statement is false. a. b. c. d. If A ;.! B and B ;.! C, then A ;.! C. If A C B and B ~ C, then A ~ C. If x E:: B and B E:: C, then x E:: C. If A E:: B and B ~ C, then A (;{. C. 3. If A is a set prove: (A C 0) <'* (A = 0) 4. Give two examples of each of the following: A E:: B, A C B, A C B, and A = B. 5. Form the power set of a set containing four elements; five elements. How many elements are there in each of the power sets described?

CA, {a} CA, {b} CA, {a, b} CA Hence, the set of all subsets of A is given by 1¢, {a), lb}. {a, bl} Similarly, if B = Ix,y,z}, then the set of all subsets of B is the set {¢, {x}, {y}, {z}, {x,y}, {x,zl, {y,z}, {x,y,zl} The set of all subsets of a given set is called the power set of that set, and we formalize this concept in the following definition. 2 • 33 DEFINITION 2 . 6 For every set A, the set of all subsets of A, denoted p(A), is called the power sel of A. * As seen from the two examples above, when a set contains two elements, its power set contains four elements.