Sets using the Venn Diagrams

When I  first was introduced to sets and whole numbers, I had no clue what I was doing. It was not until a couple of years ago when I had to take a math class to finish up my A.A. degree that I finally understood the concept of sets and whole numbers. It was Venn diagram with universal set that helped me understand the concept. 

A few things that a person needs to understand when dealing with sets and whole numbers are: What is a set, one-to-one correspondence and the difference between equal and equivalent sets:

A set is any collection or objects or ideas that can be listed or described.  For example, the set of boy names contains the names Tom, Tucker, and Doug. In mathematical shorthand, we list the objects in a set within braces :{ Tom, Tucker, Doug}.

Sets A and B have one-to-one correspondence if and only if each element of A is paired with exactly one element of B and each element of B is paired with exactly one element of A. 

A = {m, n, o, p, q}

B = {w, x, y, z, a}

 The different between equal sets and equivalent set. A= {1,2,3,4,5,6} and B = {1,2,3,4,5,6} A and B above are equal sets because all their elements are precisely the same. C= {a,b,c,d,e,f} and D= {3,4,5,6,7,8} C and D are equivalent sets because the number of elements in both the sets is the same i.e.6. 

How we come to the part that talks about Venn diagrams with universal sets. The relationship of a set to its universal set leads to an important definition of the complement of a set:

The complement of a set A, written A, consists of all of the elements in U that are not in A.

Here is the math problem from my math class that made set with whole number using a Venn diagram with universal sets that made me understand the concept of sets and whole numbers. I believe it was the use of a real-life situation that helped me understand. 

A winter resort took a poll of its 350 visitors to see which activities people enjoyed. The results were as follows: 178 liked ski, 154 liked to snowboard, 57 liked to ice skate, 49 liked to ski and snowboard, 15 liked to ski and ice skate, 2 liked to snowboard and ice skate, and 2 liked all three activities. Complete the Venn diagram below and determine the cardinality for each region

First step: The first step to filling in the four inner circles is to look at the numbers with and/all. Where the three circles meet this is where you will put all (2)

Second Step: Now that you have your all filled in, to find the next three you will subtract all from the and. Example: to find how many like both snowboarding and ice skating you would take 2 (all) from 2 giving you 0, 2-2=0. To find skiing and ice skating you subtract 15 and 2 to give you 13, 15-2=13. Follow the same for ski and snowboarding, 49-2=47

Third step: to find the individual sport lovers you will subtract 13,2,47 (the inner circles) from those that like the individual sport. Example: Skiing =178 people liked skiing, so 178-13-2-47= 116 people who just like skiing. Follow this step for the other two circles. 

Fourth step: Add all circles (including inner) to find how many people took the survey. Subtract the number from the number of visitors polled to find how many didn’t apply. Example: 116+105+42+13+2+47=325(people who took the poll.) 350-325=25 (people who didn’t apply) This number goes outside the circles into the universe.  

This is a great introduction video on Venn Diagrams.