Sets and Prepositions Discrete Mathematics Theory

Sets and Prepositions

A set is a collection of distinct objects. Thus, the group of all sophomores in the university is a set. So is the group of all Computer Science majors in the university, and so is the group of all second-year Computer Science majors. We use the notation {a, b, c} to denote the set which is the collection of the objects a, b and c. The objects in a set are also called the elements or the members of the set. We usually also give names to sets. For example, we write S = {a, b, c} to mean that the set named S is the collection of the objects a, b and c. Consequently, we can refer to the set S as well as to the set {a, b, c}.

Note that a set contains only distinct elements. Thus, {a, a, b, c} is a redundant representation of the set {a, b, c}. Similarly, {The Midnight Visitor, The Midnight Visitor, The Missing Witness, 114-Main-Street} is a redundant representation of the detective stories written by A. B. Charles.

In many cases, when the elements in a set share some common properties, we can describe the membership of the set by starting the properties that uniquely characterize the elements in the set. For example, let S = {2, 4, 6, 8, 10}. We can also specify the elements of S by saying that S is the set of all even positive integers that are not larger than 10. Indeed, we can use the notation

S = {x|x is an even positive integer not larger than 10} for the set {2, 4, 6, 8, 10}.

In general, we us the notation,

{x|x possesses certain properties} for a set of objects that share some common properties.

Thus, S {Smith, Jones, Wong, Yamamoto, Vogeli}

And, S = {x|x is a second-year Computer Science major} are two different ways to describe the same set of elements.

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