Paper Example on Set Theory

The set can be expressed as {x: x 56} or {x | x 56} in builder notation. The expression means that the set represented is a set that contains all elements x which are greater or equal to 56.

{3, 4, 5, 6, . . . ,37}

The set can be expressed as {x: 3 x 37} or the set {x | 3 x 37} in builder notation. And, this implies that all the values of x in the set are greater than or equal to 3 but less than or equal to 37.

Given the following: = {1, 2, 3, 4, 5, 6, 7, 8} A = {1, 7, 8} B = {2, 7, 8} C = {2, 3, 4, 5, 6}

A B

The intersection of set A and set B is A B = {7, 8}.

C O

The intersection of set C and the empty set is C O = O

(A B) B

The intersect of A Union B and set B is (A B) B = B

Thus, the correct answer is B = {2, 7, 8}

Two Page Report

Abstract

A set is any well-defined collection, group, aggregate, class or conglomerate of objects (Kuratowski, 2014). These objects are called elements of the set and are often said to be members of the set. A set is often specified by the following features:

Listing its element inside a pair of braces or curly brackets or parentheses

Means of a property of its elements

It is the aim of this paper to apply set theory and laws associated to sets to solve some mathematical problems. Much focus will be on the builder notations and the fundamental operations and sets IN solving the problems.

Set Theory Module 1

The activity entailed the expression of each set using builder notation and inequality notation to express conditions. For instance, condition x must be met for it to be a member of a given set. According to Kunen (2014), a set builder notation is a shorthand form used in writing sets and subsets. These notation helps in instances where the set has an infinite number of element.

{56, 57, 58, 59, . . .}

Notation {x: x 56} or the set {x | x 56}

The set {x: x 56} is the set of all x such that x is greater than or equal to 56. Can also be noted by replacing the colon sign by a vertical line as shown above.

{ 3, 4, 5, 6, . . . , 37}

Notation {x: 3 x 37} or the set {x | 3 x 37}

The set {x: 3 x 37} is the set of all x such that x is greater than or equal to 3 and x is less than or equal to 37. Can also be noted by replacing the colon sign by a vertical line as shown above.

Given,

= {1, 2, 3, 4, 5, 6, 7, 8}A = {1, 7, 8}B = {2, 7, 8}C = {2, 3, 4, 5, 6}

We find

A B

The set A B = {x: x A and x B} Since in the case provided, A = {1, 7, 8} and B = {2, 7, 8} it is thus correct to state that the intersection of A and B can be expressed as A B = {7, 8}. This is because the intersection law lists the elements that are common to two or more sets.

C O

The set O is an empty set with no elements. And, thus using the domination law of set theory, it is correct to state that the set C O = O.

(A B) B

The sets A = {1, 7, 8}, B = {2, 7, 8} therefore the set A B = {x: x A or x B},

A B = {1, 2, 7, 8}. The set (A B) B is the set {x: x (A B) and x B}

(A B) B = {2, 7, 8} = B.

By applying the absorption law of set theory, it is thus correct to state that

(A B) B = B which is B = {2, 7, 8}

Conclusion

From the task, it was noted that, for a set A, written as x A implies that x is a member of A or belongs to set A. In addition, a null set denoted as O is an empty set, a set with zero elements. Therefore, the three sets A, B, C and null or empty sets could be represented as:

A B = {x: x A or x B}

A B = {x: x A and x B}

(A B) B = B

C O = O

In conclusion, set builder notation was successfully used in solving questions 1 and 2. For question 3, 4 and 5, fundamental set operations and laws associated with set theory were used in solving the problems successfully.

References

Kunen, K. (2014). Set theory an introduction to independence proofs (Vol. 102). Elsevier.

Kuratowski, K. (2014). Introduction to set theory and topology (Vol. 101). Elsevier.