## Re-Venn-ge

Sets are a powerful tool that allow us to look at groups of items instead of dealing with each item individually. By recognising some common property of a group of items, we can define them as a set, for example all left-handed, red-headed spanish plumbers could be used as a set. Not all members of the set would be identical in all ways – they could for example be male or female – but they would all be left-handed, red-headed and spanish. We use a specific set of symbols to work with sets, just as we use a particular set of symbols to write english. The “language” of sets is very precise, and you must ensure that you use it correctly. Here are two links (1, 2) to pages that give clear instructions on set notation (the symbols used for working with sets).While working with sets algebraically, using set notation, is very powerful, it is not particularly easy to “see” what is happening sometimes. For this, we turn to Venn Diagrams. Venn diagrams are named for John Venn, a 19th century logician and philospher. Venn Diagrams are a visual representation (a picture) of what is shown in a set notation statement. Venn Diagrams are very useful because they are a shortcut; they make a set statement clear and immediately relevant. You will need to recognise four main areas in a (2 set) Venn Diagram:

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When we work with set notation in Venn Diagrams, we have one primary rule:

n(AUB) = n(A) + n(B) – n(A∩B)

This formula says to count the number of items in both sets (the union of A and B), we add the number of items in set A to the number of items in set B and take away the number of items that are in both sets (simultaneously).

Why do we have to take away the items that are in both sets? If you watch the video above again, you will see that we added everything that was in A (thats the n(A) part of the formula), not everything that was in A and not in B, thus when we added everything that was in B, we were counting those items that are in both A and B a second time (twice!), so we have to take them out (that’s the – n(A∩B) part of the formula).

See you in class!