D.1: Set Theory
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We all have some intuitive understanding of the notion of a ''set''. However, the notion of set is know in mathematics as ''primitive''; that is, it is a basic term that is commonly understood but is extremely difficult to define in a way that would be acceptable to mathematicians (other primitives include point, line, and plane).<br> | We all have some intuitive understanding of the notion of a ''set''. However, the notion of set is know in mathematics as ''primitive''; that is, it is a basic term that is commonly understood but is extremely difficult to define in a way that would be acceptable to mathematicians (other primitives include point, line, and plane).<br> | ||
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| + | Informally then a set is a collection of objects, real or imagined. Generally, these objects have one or more common properties that separate them from other objects. An example of a set would be '''A''' = {banana, lemon, strawberry, orange, grape}. Clearly, all the objects of this set are fruits. This is an example of a set of real objects, but a set can just as well be made up of abstractions. Let '''B''' = {democracy, republic, communistic, dictatorship}. '''B''' is a set of forms of government. The elements of a set as all the things that belong in that set. In the first example the elements are banana, lemon, strawberry, orange, and grape.If we denote the name of a set with a capital letter, then the elements are denoted with the lower-case letter with subscript to give the elements distinction but not order. Thus, '''A''' is the given set of fruits, then a<sub>1</sub> = banana, a<sub>2</sub> = lemon, a<sub>3</sub> = strawberry, a<sub>4</sub> = orange, a<sub>5</sub> = grape. We could have just as readily said a<sub>3</sub> = banana since there is no order to the set. | ||
| - | + | <br>Two sets are called equal if and only if every element in one set exactly matches another element in the second set for all elements in both sets. Mathematically we say given '''A''' and '''B''' are sets, then '''A''' = '''B''' iff '''a''' = '''b''' for all '''a''' in '''A''' and all '''b''' in '''B'''. Notice that the definition requires that both sets be the same size; that is, they both have the same number of elements. | |
| - | + | <br>If we look at the set A above of fruits, there are several ways that we can group elements into smaller groups. For example, within the set there are two fruits that are yellow when ripe: banana and lemon. Alternately, there are two fruits that are generally called citrus: lemon and orange. Both of these are examples of what are called a subset. Mathematically, a subset of a set, denoted A | |
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| + | For further information about basic set concepts or a more rigorous treatment, refer to the following: | ||
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| + | people.umass.edu/partee/NZ_2006/Set%20Theory%20Basics.pdf | ||
Revision as of 16:11, 22 July 2010
Discrete Math Learning modules
Oregon Department of Education standards for Set Theory:
D.1.1 Demonstrate understanding of the definitions of set equality, subset and null set.
D.1.2 Perform set operations such as union and intersection, difference, and complement.
D.1.3 Use Venn diagrams to explore relationships and patterns, and to make arguments about relationships between sets.
D.1.4 Demonstrate the ability to create the cross-product or set-theoretic product of two sets.
Sets
We all have some intuitive understanding of the notion of a set. However, the notion of set is know in mathematics as primitive; that is, it is a basic term that is commonly understood but is extremely difficult to define in a way that would be acceptable to mathematicians (other primitives include point, line, and plane).
Informally then a set is a collection of objects, real or imagined. Generally, these objects have one or more common properties that separate them from other objects. An example of a set would be A = {banana, lemon, strawberry, orange, grape}. Clearly, all the objects of this set are fruits. This is an example of a set of real objects, but a set can just as well be made up of abstractions. Let B = {democracy, republic, communistic, dictatorship}. B is a set of forms of government. The elements of a set as all the things that belong in that set. In the first example the elements are banana, lemon, strawberry, orange, and grape.If we denote the name of a set with a capital letter, then the elements are denoted with the lower-case letter with subscript to give the elements distinction but not order. Thus, A is the given set of fruits, then a1 = banana, a2 = lemon, a3 = strawberry, a4 = orange, a5 = grape. We could have just as readily said a3 = banana since there is no order to the set.
Two sets are called equal if and only if every element in one set exactly matches another element in the second set for all elements in both sets. Mathematically we say given A and B are sets, then A = B iff a = b for all a in A and all b in B. Notice that the definition requires that both sets be the same size; that is, they both have the same number of elements.
If we look at the set A above of fruits, there are several ways that we can group elements into smaller groups. For example, within the set there are two fruits that are yellow when ripe: banana and lemon. Alternately, there are two fruits that are generally called citrus: lemon and orange. Both of these are examples of what are called a subset. Mathematically, a subset of a set, denoted A
For further information about basic set concepts or a more rigorous treatment, refer to the following:
people.umass.edu/partee/NZ_2006/Set%20Theory%20Basics.pdf
