Survey of Math - MAT 140

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Set Concepts Sets are collections of "things" that are called elements. We looked at sets in our discussion of the Real Numbers, such as the Counting Numbers f1; 2; 3; 4; 5; :::g. Sets can be a collection of any kinds of "things" that we will call Elements.

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Roster Form

Example 1 The planets is the set of elements that include Mercury, Venus, Earth, Mars, Jupiter, Saturn, Neptune, Uranus, and Pluto. We usually write this in what is called ROSTER FORM. And we can name the set using a single letter, P=planets.

P D fMer cur y; V enus; Ear th; Mar s; J upiter; Satur n; N eptune; Uranus; Plutog To be able to understand these better it is best to read each symbol with its English equivalent. P "the set of Planets"

D "is"

f "the set of elements"

Mer cur y; :::; Plutog

Roster Form lists all elements of a set, or gives an indication of a continuing pattern, like the counting numbers. Example 2 Roster the set of Weekdays W : W D fMonday; T uesday; W ednesday; T hur sday; Fridayg Example 3 Roster the set of Counting Numbers .A/ that are between 7 and 12 A D f8; 9; 10; 11g Notice here that the numbers 7&12 do not appear. That is because of the word "between"

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Set Builder Notation

Another type of notation often used when the Roster Form list will be too long or hard to list using any patterns is Set Builder Notation. A D f2; 3; 4; 5; 6; 7; 8; 9; 10g is in Roster Form A D fn 2 Nj 2

n

10g is in Set Builder Notation

In English the Set Builder Notation reads: A "the set A" D "is" f "the set" n 2 N "the variables are natural numbers" j "such that" 2 2 and 10 inclusively" Copyright 2007 by Tom Killoran

n

10 "it is between

Survey of Math - MAT 140

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Example 4 Roster the Set of Weekdays .W / and then write this set using Set Builder Notation W D fMonday; T uesday; W ednesday; T hur sday; Fridayg W D fx 2 Daysj x is a weekdayg

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Well De ned Sets

We have to be careful with these sets and the way that we de ne them, for instance: D D fx 2 actor sj x are good looking bald actorsg This is not a "Well De ned Set" . It is possible to include different actors, depending on who you ask. We need these sets to be the same for all audiences. D D fx 2 actor sj x are actors that won An Oscar g is a well de ned set. Example 5 Is the set of "summer months" a well de ned set? Most people would say the summer months would be June, July, August. But Summer doesn't start until near the end of June and goes to the near the end of September! So in and of itself, summer months is Not Well De ned!

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Equal Sets

A set A is said to be EQUAL to a set B if and only if they have the exact same elements (and the same number of elements). Example 6 Given:

A D fapples; oranges; pear sg and

B D fx 2 f oodjx is a fruitg

Is A D B? Why or Why not? No. Even though every element of A is also an element of B; we know that B has more elements than A: Mainly "banana" or "plum".

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Cardinal Number

The Cardinal Number of a Set is the number of elements in a set. Example 7 What is the Cardinal Number of the Set A D fapples; oranges; pear sg? The Cardinal Number of this set is 3 since there are three elements The Question could have used the notation: n .A/ D 3 where n is read, the "Cardinal Number of...." Copyright 2007 by Tom Killoran

Survey of Math - MAT 140

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Equivalent Sets

Equivalent Sets are any two sets that have the same cardinal number (same number of elements) Example 8 Given

A D f1; 2; 3; 4g and

B D fx N j5

x

8g can we determine if A and B equivalent?

Since Equivalent sets depends upon their Cardinal Numbers we nd: n .A/ D 4 and n .B/ D n .f5; 6; 7; 8g/ D 4 so yes they are equivalent and we write: A B (the three line equal sign is read equivalent) Equivalent sets are also known as one-to-one sets. For every element of one set there is a corresponding element in the other set.

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Null or Empty Set

Denoted as ? "null" or as fg "empty" . Both mean the same thing, a set that contains no elements or a set whose Cardinal Number is Zero! Example 9 Given A D fx 2 star sjx is a planetg we can deduce that the set in Roster form would have to be A D fg since there are no stars that are also planets.

NOTE: ? D fg or the Null Set is Equal to the Empty Set

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Universal Set

The universal set is used to limit the number of Ideas a set can have. It de nes a conversation of sets to a speci c topic, of what we are taking about. If we had sets based on the State of the US, then the universe would contain all of the States and nothing else. If we had sets based on numbers less than 100 then the Universe would contain only numbers from 1 to 99, and nothing else.

Copyright 2007 by Tom Killoran

Survey of Math - MAT 140

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Questions: True or False / Explain Why or Why Not... 1. d 2 fa; b; c; e; f; gg

2. apples 2 fx is food j x is a vegetableg

3. f5g 62 f1; 2; 3; 4; 5g

4. 8 2 fx 2 N jx < 8 g

5. f|; }; ~; •g is equal to fx 2 N jx < 5 g

6. f|; }; ~; •g is equivalent to fx 2 N jx < 5 g

Copyright 2007 by Tom Killoran

Survey of Math - MAT 140

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Answers: 1. False, the element d is not an element of the set.

2. False, apples are not of the set of vegetables. They are a fruit

3. False, the "Set of 5" is not an element of f1; 2; 3; 4; 5g : To be an element we would need to replace the 5 with f5g:

4. False, 8 is not in the set of Natural Numbers less than 8

5. False, The Cardinal number of f|; }; ~; •g (which is 4) is the same as fx 2 N jx < 5 g , but the elements are not the same.

6. True, The Cardinal number of f|; }; ~; •g is the same as fx 2 N jx < 5 g.

Copyright 2007 by Tom Killoran