Graphing Cartesian Products and Unions: Set Theory Sketches

[cubicmonkey]

So the book asks me to sketch out these graphs, and of course there are no examples. I was wondering how this is done.

(a) [0,1] X [1, 2] // The X here stands for the Cartesian product.

(b) ([0,1] U {2}) X [1,2] // How can I graph this? The U stands for Union and the X here stands for the Cartesian Product.

(c) ([0,1] U {2}) X ([1,2] U {3}).// Again, the U stands for Union and the X here stands for the Cartesian Product.

 

[dodo]

I imagine they mean rectangular areas in the $\mathbb{R}^2$ plane. For the (a) example, set one interval in the X-axis and the other in the Y-axis; this defines a rectagular area (the points which x is in [0,1] and which y is in [1,2]).

 

[cubicmonkey]

Thanks, DODO that was what I suspected for part a, but parts b and c still mystify me.

 

[cubicmonkey]

([0,1] U {2})

For example what would this look like?

 

[chiro]

([0,1] U {2})

For example what would this look like?

Think of your Venn diagrams. If you have two sets A and B and you have A U B then you can have anything that is both A and B. If you have A $\bigcap$ B then you have any element that is both A and B. This is the definition of these two binary operators.

Again if you get stick think of the Venn diagram graphically of what A $\bigcup$ B and A $\bigcap$ B in terms of pictures and then use that intuition to think of what the symbols mean.

Simplify anything with $\bigcup$ and $\bigcap$ and then take the cartesian product after.

 

[cubicmonkey]

So I uploaded my guess at it. What do you think? Is (b) correct? Does ([0,1] U {2}) simply become [0,1,2], which could be read as [0,2]?

 

[chiro]

So I uploaded my guess at it. What do you think? Is (b) correct? Does ([0,1] U {2}) simply become [0,1,2], which could be read as [0,2]?

You have to be careful with your notation.

Usually when we want to describe a discrete (countable) set, we usually specify every element in the set and not just the first and last element. When we are talking about a continuous set like all real numbers from 0 to 2 inclusive then we say [0,2]. It is probably a better idea to specify your set as [0,1,2] just so there is no confusion. Your answer is right of course but your [0,2] to mean {0,1,2} is misleading: (also when we talk about sets we always put them in curly braces like {0,1,2}: [0,2] is usually used for describing intervals like 0 <= x <= 2)

So for the sets {0,1} U {2} = {0,1,2} remember to use the curly braces just so no-one gets confused :)