© 2008  Rasmus ehf    og Jóhann Ísak

Sets I

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Lesson 1

  Set Theory and notation 

Any  collection of numbers, objects or ideas e.t.c. is called a SET and each object in  the set is called an  element of the set. Most sets that we deal with are sets of numbers and each number belonging to that set is called an ELEMENT of the set.

Sets are written using curly brackets that contain the elements of the set. For example the set containing only the numbers 1, 2 and 3 tis written {1,2,3}.

The even numbers greater than 0 can be written {2,4,6,8,∙∙∙∙∙∙∙}. The dots indicate that the numbers continue following the same pattern.

Mathematics has a whole language of symbols that are used to represent words and statements. This enables us to state precisely and logically what we mean without using a lot of words. Many of these symbols are used in set theory. Here are some examples:

The symbol means “belongs to” or “is an element of”.

If we put  a line through this symbol i.e. cross it out, , then it means “does not belong to” or “ is not an element of”.

If M = {0,1,2,3} then we can make the statements

{1} M   and   {4} M

Which are both true.

Sometimes it’s easier to describe a set rather than write a list of it’s elements . A vertical line | inside the set brackets is the symbol for “such that” or “for which it is true that “.  For example the set of numbers belonging to the set M that are less than 3 can be written

{x M | x < 3 } = {0, 1, 2}

This reads “x is an element of the set M such that x is less than 3”.This is true of the numbers 0, 1 og 2.

We can also describe the above set as being the same set as M except for the number 3. A backslash \ is the symbol  used to  mean “except for”. Using this notation we can write:

We often need more than one condition to describe a set. Here’s an example:

{x M | x < 3  og x 0} = {1, 2}

This reads “ x is an element of the set M such that x is less than 3 but not equal to 0.

If we have a condition that no elements satisfy we have a set called the empty set. In other words a set with nothing in it.

The symbol Ø or { } is used for the empty set.  For example

{x M | x > 3 } = Ø

This reads: the set of all the numbers in M that are greater than3. As M contains only the numbers 0, 1, 2, 3 there are no numbers greater than 3.

When we look at the sets A = { 1,2,3,4,5 }, B = { 1,3,5 } and C = { 2,4,6 } we can see at once that all the elements in the set B are also elements of the set A. We say that B is a proper subset of A.  The symbol is used to express this relation. We write   

B A

Notice also that the set C has an element, 6, that is not in the set A. In this case C is not a subset of A. The symbol for this is made by puttting a line through , in fact we cross it out as we have done with other symbols when something is not true. In set notation we write  

 C A

We use the word “proper subset” if one set is contained in another but they are not the same set. If one set is contained in another but could be the same set we use the word subset and the symbol . This is comparable to the use of the inequality symbols in algebra, < and .

A list of symbols and their meanings

Try Quiz 1 on Sets 1.  
Remember to use the checklist to keep track of your work.