Concept of Sets

Introduction to Sets

A set is a group of objects referred to as elements of the set. Sets of numbers are used in basic mathematics such as in geometry and trigonometry to evaluate equations. A mathematical equation is a function that maps elements from an original set to elements in an answer set. 

The first set is the function's domain. It is the group of numbers for a particular function that is defined by mathematical laws or outlined by definition. If it is undefined the domain is usually assumed to be the set of all real numbers. 

The answer set is the function's range. The range is defined as the set of all possible numbers that are answers to the function. 

The empty set is the set with no elements at all.

Universal Sets

The universal set is the set of all the elements acceptable to a particular mathematical function. The universal set contains all possible numbers that work with the function. Normally, the universal set is the set of real numbers. It is equivalent to the unrestricted domain of the function. The universal set for the function is restricted from any number that yields an impossible result, such as an imaginary or infinite number. 

The square root function restricts the use of negative numbers, so its universal set is the set of all real numbers greater or equal to zero. 

A complement to a set is the difference between the original set and the universal set. It is a proper subset of the universal set. A proper subset is any set contained entirely within the universal set, but is not equivalent to it. Every element must be in the universal set. The power set is a special set that contains all the proper subsets of a set. A proper subset can have an infinite number of elements. For example, the set of integers is a well-defined infinite set that is a proper subset of the set of real numbers which is also infinite.

Examples

In the mathematical equation x + 1 the universal set is the set of all real numbers.

In the mathematical equation the square root of x the function restricts the universal set to the set of all real numbers greater than zero, because the square root of a negative number is undefined or an imaginary number.

Power Set

A power set is a specially defined set using the elements of another set. The power set contains all possible combinations of elements from the original set. So, the power set contains all the proper subsets of a given set, the set itself and the empty set as its elements. 

The word power refers to the mathematical power function. The number of elements in the power set is always equal to the number of elements in the original set to the power of two. The power set determines all possible set combinations of a given set. 

Every set has a power set, but a set with an infinite number of elements will have a power set with an infinite number of elements. In other words, the power set of an infinite set is infinite.

Examples

Using the set of numbers (1, 2, 3, 4) the elements of this power set are all the proper subsets (1, 2, 3, 4), (1, 2, 3), (1, 2, 4), (1, 3, 4), (2, 3, 4), (1, 2), (1, 3), (1, 4), (2, 3),(2, 4), (3, 4), (1), (2), (3), (4) and (the empty set). 

There are sixteen elements in this power set. This is equal to four squared, which is four to the power of two.

Set Union

The union of two sets consists of all the elements that are in both sets. Any elements in both the originating sets are single elements in the union set. If the sets are disjointed, where there are no elements in common, then the number of elements in the union can be found by the addition of the total number of elements. 

The union of more than two sets is the union of two sets taken as a union with the third set and so forth. 

The associative law and the commutative law for sets states that it does not matter which two sets are taken first or in what order they are taken. 

The union of two sets is different from the intersection of two sets in that the element has to be in both sets for it to be in the set intersection.

Examples

The union of the set of numbers (1, 2, 3, 4) and the set of numbers (3, 4, 5, 6) is the union set (1, 2, 3, 4, 5, 6).

The union of the set of numbers (1, 2, 3, 4) and the set (5, 6, 7, 8) is the union set (1, 2, 3, 4, 5, 6, 7, 8).

The union of the set of numbers (1, 2, 3, 4) and the set (1, 2) is the union set (1, 2, 3, 4).

Set Intersection

The intersection of two sets consists of all of the elements that the two sets have in common. The intersection set is a proper subset of both the originating sets. If the sets are disjointed sets then there are no elements in common between them and the intersection is the empty set. 

The intersection of more than two sets can be found by intersecting two sets and then intersecting the resulting set with the third set and so forth. 

The associative law and the commutative law for sets state that it does not matter which two sets are taken first or in what order they are taken. 

The intersection of two sets is different from the union of two sets in that either set can contain the element for it to be in the set union.

Examples

The intersection of the set of numbers (1, 2, 3, 4) and the set of numbers (3, 4, 5, 6) is the set (3, 4).

The intersection of the set of numbers (1, 2, 3, 4) and the set of numbers (5, 6, 7, 8) is (the empty set).

Set Complement

The complement of a set is a set of elements not in the set, but in the universal set. 

By definition all the original set's elements must be part of the universal set. The original set and its complement are both proper subsets of the universal set. Every element in the complementary set is in the universal set, and every element in the original set is in the universal set. No element in the original set is in the complementary set. 

The number of elements in the complementary set is the difference between the number of elements in the universal set minus the number of elements in the original set.

Examples

The complement of the set of numbers (2, 3, 4) if the universal set is defined as the set of numbers (1, 2, 3, 4, 5, 6, 7, 8, 9, 10) is the set (1, 5, 6, 7, 8, 9, 10).

The complement of the set of numbers (1, 2, 3, 4) and the set of natural numbers is the set (5, 6, 7, 8 . . .).

The Identity Laws

The identity laws establish the basic rules for taking the union and intersection of sets including the empty set. They apply to all sets including the set of real numbers.

Where A is any set of numbers:

1) A union A equals A 

This law states that the union of two identical sets is the same as the original set. 

2) A intersection A equals A 

This law states that the intersection of two identical sets is the same as the original set. 

3) A union empty set equals A

This law states that the union of a set and the empty set is the same as the original set. 

4) A intersection empty set equals empty set 

This law states that the intersection of a set and the empty set is the same as the empty set.

The Commutative Laws

The commutative laws establish the rules to the order of the sets when taking the union and intersection. They apply to all sets including the set of real numbers.

Where A and B are sets of numbers:

A U B = B U A 

A union B equals B union A

This law states that the union of two sets is the same no matter what the order is in the equation.

AB = B A

A intersection B equals B intersection A

This law states that the intersection of two sets is the same no matter what the order is in the equation.

The Associative Laws

The associative laws establish the rules of taking unions and intersections of sets. They apply to all sets including the set of real numbers.

Where A, B and C are sets of numbers:

A U (B U C) = (A U B) U C

A union (B union C) equals (A union B) union C

This law states that taking the union of a set to the union of two other sets is the same as taking the union of the original set and one of the other two sets, and then taking the union of the results with the last set.

A(BC) = (AB)C

A intersection (B intersection C) equals (A intersection B) intersection C

This law states that taking the intersection of a set to the intersection of two other sets is the same as taking the intersection of the original set and one of the other two sets, and then taking the intersection of the results with the last set.

The Distributive Laws

The distributive laws establish the rules of taking unions and intersections of sets. They apply to all sets including the set of real numbers.

Where A, B and C are sets of numbers:

A U (BC) = (A U B)(A U C)

A union (B intersection C) equals (A union B) intersection (A union C)

This law states that taking the union of a set to the intersection of two other sets is the same as taking the union of the original set and both the other two sets separately, and then taking the intersection of the results. 

A(B U C) = (AB) U (AC)

A intersection (B union C) equals (A intersection B) union (A intersection C)

This law states that taking the intersection of a set to the union of two other sets is the same as taking the intersection of the original set and both the other two sets separately, and then taking the union of the results.

The DeMorgan Laws

The DeMorgan laws establish the rules of taking complements of sets. They apply to all sets including the set of real numbers.

Where A and B are sets of numbers:

C (A U B) = C (A) C (B) 

The complement of (A union B) equals the complement of (A) intersected with the complement of (B)

This law states that the complement of the union of two sets is the intersection of the complements. 

C (AB) = C (A) U C (B)

The complement of (A intersection B) equals complement of (A) united with the complement of (B)

This law states that the complement of the intersection of two sets is the union of the complements.