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Sets are used much of the time in geometry and although some aspects of Euclidean geometry, which is the regular geometry most people are familiar with, can be described without sets, it can be helpful to use them to describe certain geometric objects and their relationships. In geometry, a set consists of a group of objects that are counted as a unit. If the number of objects is unlimited then the set is infinite. In math, the set of the whole numbers consists of 0, 1,2,3 and so on to infinity. This would be an example of an infinite set. The objects in a set are called elements or members. If they can be limited to a specific number, it is a finite set. ## Types of Sets## SubsetA set can also be formed from the elements of another set. If some or all of the elements are included in the new set, it is called a subset. An example of a subset would be if set a contained 1,2,3,4 and 5 and the subset contained 1, 2 and 3.
The symbol is used to indicate that a set is a subset. From the example above where B is the subset of A, the following can be written: B A A subset or set is indicated, sometimes, by the symbols { and } which contain its elements or members. ## SupersetA superset is the larger set that a subset is taken from. For example, given the two previous sets A = {1, 2, 3, 4, 5} and B = {1, 2, 3}, A would be the superset of B. ## Empty SetThe empty set is a set with no members or elements. It is also called the null set and is symbolized by {} or . ## Power SetThe power set is a special kind of set that includes a set as an element along with all of its subsets including the empty set. If A = {1, 2, 3}, then the power set of A would be the following set B: B = { {1, 2, 3}, {1, 2}, {1, 3}, {2, 3}, {1}, {2}, {3}, {} }
The number of elements in a power set is 2 ## Geometric Examples of SetsIn geometry, a line can be thought of as an infinite set of points where the set itself points only in two opposite directions. A plane can also be thought of as a set containing an infinite number of points. In geometry, a plane is usually represented by the following diagram. A circle can be described as the set of all points in a plane that are the same distance from another point in the same plane. An arc on the circle is a subset since it only contains some of points in the set that is the circle. A line segment is another example of a subset. It only contains some of the points that are on the line that travels through it. ## Set Operations## UnionSets can be joined together in such a way that all of the members of one set are counted along with the members of the other one or more sets. Consider, for example, the following sets:
If these sets are joined together so that all the members are present, it would create a new set:
When sets are joined together in this way, it is called a union. The symbol for a union is . In the example above, the union of sets A and B can be written as: A B Since the result of the union is set C, it can also be written as: C = A B Another less concise way of writing this would be: {1,2,3,4,5} = {1,2,3} {4,5,6} ## IntersectionTo select the matching or identical elements between sets, the intersection is used. In the previous example, A = {1,2,3} and B = {3,4,5}. The element that both of these sets have in common is the 3. The intersection of sets A and B is written as follows: A B is the symbol for intersection. If C is the resulting set with the common elements of A and B, it is written as: C = A B It can also be written as: {3} = {1,2,3} {4,5,6} ## ComplementSometimes, a set can be formed by selecting the elements from a first set that are not members in a second set even if the first set includes elements of the second set. These selected elements would form a third set called the complement of the second set. In other words, the complement of a set is all the elements not in the set. The complement, however, is usually applied to the subset of a larger set to get another subset with different members. For example, if A = {1, 2, 3, 4, 5, 6, 7} and B = {1, 2, 3}then the complement of B would be C = {4, 5, 6, 7}. The symbol used to indicate a complement is usually an overline or a superscript c such as the following. If A is a set then A would be the complement of A.
If B is a set then B ## Geometry Examples of Set OperationsA triangle can be thought of as a union of at least three line segments. A line can form by taking the intersection of two planes. A point can also be described as the intersection of two lines. An angle can be represented as the union of two rays: ## Venn DiagramsAnother way of describing sets is with Venn diagrams.
It starts with the universal set called
The subsets in The union of two or more sets would include the interiors of all the circles in the union minus the overlaps to remove duplication: The intersection would be the overlapping region of the two sets. |

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