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Geometry: Sets and Set Theory

 



Set Theory
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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

Subset

A 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.

A = {1, 2, 3, 4, 5} where A would be the set.
B = {1, 2, 3} where B would be the subset.

The Subset symbol 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 Subset symbol A

A subset or set is indicated, sometimes, by the symbols { and } which contain its elements or members.

Superset

A 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 Set

The empty set is a set with no members or elements. It is also called the null set and is symbolized by {} or Symbol for the empty set or null set.

Power Set

The 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 2n where n is the number of elements in the original set. In the case above, the number of elements in the original set, A, is three while the number of elements in the power set is 23 = 8.

Geometric Examples of Sets

In geometry, a line can be thought of as an infinite set of points where the set itself points only in two opposite directions.

A line is a set containing an infinite set of points



A plane can also be thought of as a set containing an infinite number of points.

A plane is a set containing an infinite set of points




In geometry, a plane is usually represented by the following diagram.

A geometry representation of a plane as an infinite set of points



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.

A circle is a set of points with each point an equal distance from a center point

An arc on the circle is a subset since it only contains some of points in the set that is the circle.

An arc is a continuous subset of points on a 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.

A line segment is a continuous subset of points on a line

Set Operations

Union

Sets 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:

A = {1, 2, 3}
B = {3, 4, 5}

If these sets are joined together so that all the members are present, it would create a new set:

C = {1, 2, 3, 4, 5}

When sets are joined together in this way, it is called a union. The symbol for a union is Union symbol. In the example above, the union of sets A and B can be written as:

A Union symbol B

Since the result of the union is set C, it can also be written as:

C = A Union symbol B

Another less concise way of writing this would be:

{1,2,3,4,5} = {1,2,3} Union symbol {4,5,6}

Intersection

To 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 Intersection symbol B

Intersection symbol 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 Intersection symbol B

It can also be written as:

{3} = {1,2,3} Intersection symbol {4,5,6}

Complement

Sometimes, 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 Bc would be the complement of B.

Geometry Examples of Set Operations

A triangle can be thought of as a union of at least three line segments.

A triangle is a union of three line segments

A line can form by taking the intersection of two planes.

The intersection of two planes is a line Geometry diagram of the intersection of two planes forming a line

A point can also be described as the intersection of two lines.

The intersection of two lines is a point

An angle can be represented as the union of two rays:

An angle is the union of two rays at their common endpoint

Venn Diagrams

Another way of describing sets is with Venn diagrams.

It starts with the universal set called U that contains everything under consideration. It is usually represented by a large quadrilateral such as a rectangle or parallelogram.

Venn diagram of the universal set

The subsets in U usually just called sets and are usually represented by circles That are also labeled with capital letters. Example,

Venn diagram of two subsets as circles in the universal set

The union of two or more sets would include the interiors of all the circles in the union minus the overlaps to remove duplication:

Venn diagram of the union of two sets in the universal set

The intersection would be the overlapping region of the two sets.

Venn diagram of the intersection of two sets in the universal set
 




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