D.1: Set Theory
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= Sets<br> = | = Sets<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> | + | 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> |
| - | Informally then a set is a collection of objects, real or imagined. Generally, these objects have one or more common | + | 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. |
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| + | 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 | ||
Revision as of 15:39, 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.
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
