Math tips, Math formulas

Concept of Sets

The word "set" implies a collection or grouping of similar objects or symbols. The objects in a set have at lea& one characteristic in common, such as similarity of appearance or purpose. A set of tools would be an example of a group of objects not necessarily similar in appearance but similar in purpose. The objects or symbols in a set are called members or



Element of Set



The elements of a mathematical ret are usually symbols, ouch as numerals , lines, or points. For example, the integer6 greater than zero and less than 6 form a set, as follows:



{1, 2, 3, 4}



Notice that braces are used to indicate sets. This is often done where the elements of the set are not too numerous, Since the elements of the set (2, 4, 6) are the same as the elements of (4, 2, 6}, there two seta are said to be equal. In other words, equality between sets has nothing to do with the order in which the elements are arranged. Further- more, repeated elements are not necessary. That is, the elements of (2, 2, 3, 4) are simply 2, 3, and 4. Therefore the sets (2, 3, 4) and (2, 2, 3,4) are equal. Practice problems:



1. Use the correct symbols to designate the set of odd positive integers greater than 0 and less than 10.

of names of days of the week which do not contain the letter "s".

3. List the elements of the set of natural numbers greater than 15 and less than 20.

4. Suppose that we have sets as follows:



A = (1, 2, 3) C = (1, 2, 3, 4)



B = (1, 2, 2, 3) D = (1, 1, 2, 3)



Which of these sets are equal?



Answers:



1. 1, 3, 5, 7, 9

2. {Monday, Friday}

3. 16, 17, 18, and 19

4. A=B=D



SUBSETS



Since it is inconvenient to enumerate all of the elements of a set each time the set is mentioned, sets are often designated by a letter. For example, we could let S represent the set of all integers greater than 0 and less than 10. In symbols, this relationship could be stated as follows:



s = (1, 2, 3, 4, 5, 6, 7, 8,9)



Now suppose that we have another set, T, which comprises all positive even integers less than 10. This set is then defined as follows:



T - (2, 4, 6, 8)



Notice that every element of T is also an element of S. This establishes the SUBSET relationship; T is said to be a subset of 9.