245 OL test

The above image is a Venn diagram using only one set and the image below is a Venn diagram using two sets, [latex]A[/latex] and [latex]B.[/latex]

Study the Venn diagrams throughout this section. It takes a whole lot of practice to shade or identify regions of Venn diagrams. Be advised that it may be necessary to shade several practice diagrams along the way before you get to the final result.

We shade Venn diagrams to represent sets. We will be doing some very easy, basic Venn diagrams as well as several involved and complicated Venn diagrams.

To find the intersection of two sets, you might try shading one region in a given direction and another region in a different direction. Then you would look where those shadings overlap. That overlap would be the intersection.

For example, to visualize [latex]Acap B[/latex], shade [latex]A[/latex] with horizontal lines and [latex]B[/latex] with vertical lines. Then the overlap is [latex]Acap B[/latex]. The diagram on the left would be a first step in getting the answer. The shaded part on the diagram to the right shows the final answer.

Here are two problems for you to try. Only shade in the final answer for each exercise.

Exercise 1

1. Shade the region that represents [latex]Acap C.[/latex]
   
   Solution
   

   
   

    

2. Shade the region that represents [latex]Bcap C.[/latex]
   
   Solution

   
   

    

To shade the union of two sets, shade each region completely, or shade both regions in the same direction. Thus, to find the union of [latex]A[/latex] and [latex]B[/latex], shade all of [latex]A[/latex] and all of [latex]B.[/latex]

The final answer is represented by the shaded area in the diagram.

Here are two problems for you to try. Only shade in the final answer for each exercise.

For the complement of a region, shade everything outside the given region. You can think of it as shading everything except that region. In the Venn diagram below, the shaded area represents [latex]A^c[/latex].

Many people are confused about what part of the Venn diagram represents the universe, [latex]U[/latex]. The universe is the entire Venn diagram, including the sets [latex]A[/latex], [latex]B[/latex], and [latex]C[/latex]. The following three Venn diagrams illustrate the differences between [latex]U[/latex], [latex](A cup B cup C)^c[/latex], and [latex]U^c[/latex]. Carefully note these differences.

[latex]U[/latex]	[latex](Acup Bcup C)^c[/latex]	[latex]U^c[/latex]

Usually, parentheses are necessary to indicate which operation needs to be done first. If there is only a union or intersection involved, this isn’t necessary, as in [latex](A cup B cup C)^c[/latex] above. Convince yourself that [latex]Big((A cup B) cup CBig)=Big(Acup(Bcup C)Big).[/latex] Similarly, convince yourself of the analogous fact for intersection by performing the following steps. On the first Venn diagram below, shade [latex]Acap B[/latex] with horizontal lines and shade [latex]C[/latex] with vertical lines. Then, the overlap is [latex]Big((A cap B) cap CBig).[/latex] On the second Venn diagram, shade [latex]A[/latex] with lines slanting to the right and [latex]Bcap C[/latex] with lines slanting to the left. Then the overlap is [latex]Big(Acup(Bcup C)Big).[/latex] Check to see that the final answer, the overlap in this case, is the same for both. Shade the final answer in the third Venn diagram.