Union, Intersection and the Empty Set

Union and Intersection
The union is on the left and the intersection is on the right.

I discussed empty sets in a previous article. I hadn’t yet listened to the recent podcast of The Math Dude, Jason Marshall, in which he introduced sets and subsets. Such a curious coincidence; I thought I would continue down our united path.

This week, the sweet Math Dude talked about union and intersection of sets. So I’m heading there too – and incorporating the empty set. (Note: I’ve refrained from listening to that podcast until this article is written and published. *excited giggle*)

Union and intersection have symbols.

These symbols represent “operations” (like addition or multiplication) that you do to a pair of sets. They are called the cup and cap and look like, well, a cup and a cap. Or a U and an upside down U. The Union is the U and the intersection is the other one – that’s how I remember it and teach it.

For the sake of illustration, I’ll use Daughter’s handy SnackTraps like in the picture above.

The union is everything.

The sets I’m using are

L ={green cube, blue half circle thingie}

R = {green cube, red triangular prism, orange rectangular prism}

I’m using the letters L and R for left and right in the picture.

The union of the two sets results in all the pieces from each set, all crammed together. (And if there is a repeated item, you only take it once.)

The union of the two sets is everything combined (without duplicates).

Let’s take the set of all states that border Texas and the set of all states that border Oklahoma, like this:

T = {Louisiana, Arkansas, Oklahoma, New Mexico}

O = {Arkansas, Colorado, Kansas, Missouri, New Mexico, Texas}

Then the union of the two would be “The set of all states that border both Texas or Oklahoma or both” and would be T U O = {Louisiana, Arkansas, Oklahoma, New Mexico, Colorado, Kansas, Missouri, Texas}

Notice we don’t keep duplicates.

The intersection is only the common stuff.

Using the same L and R sets from above, I can take the intersection.

The intersection is only the stuff they have in common (we get rid of the duplicates here, too).

Let’s go back to our set of all states that border Texas and Oklahoma:

T = {Louisiana, Arkansas, Oklahoma, New Mexico}

O = {Arkansas, Colorado, Kansas, Missouri, New Mexico, Texas}

Then the intersection of the two would be “The set of all states that border both Texas and Oklahoma, at the same time” and would be T ∩ O = {Arkansas, New Mexico}.

What happens with the empty set?

When you start throwing the empty set into the mix, you follow the same rules.

For the union, you throw everything in both sets into one bag. Well, since there’s nothing in the empty set, “everything” is the original other set!

The union of the empty set with anything... is that same set back!

For the intersection, you take only the stuff that’s in common between the two. Since the empty set has nothing – there’s nothing in common.

The intersection of the empty set with any other set it just the empty set - there's nothing in common at all!

Think about it…

Does this remind you of anything? What similarities do you see between this and addition or multiplication (or subtraction or division)? Ask your children these questions too.

Get out some snack traps and blocks (or a map) and go for it! Let me know how it goes in the comments.



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5 Responses to Union, Intersection and the Empty Set

  1. In the paragraph “What happens with the empty set?”. I like the idea that two cups, one empty and the other with some objects in it, have nothing in common, hence the empty set as intersection!

    What puzzled me a bit is what if we try to find the intersection of two cups with all different objects in them? The result is again an empty set. Moreover, if the question arises, what are the ALL subsets of a given set, it will be {T, S, R}, {T, S}, {T, R}, {R, S}, {T}, {R}, {S}, O (empty set). Here I used your example Square, Triangle, Rectangle as a shorthand. We can see that power of set (set of all subsets) is exactly 2^3 = 8 i.e. 2^(# of elements) and this includes empty set too!

    In other words it can also be said that empty set is common element for empty set and any other set, which looks like a different statement that empty set has nothing in common with a set of objects. Actually, empty set is a subset of any set. If we empty the cup we will see that there is an empty set right there as well and that it is common with the other (empty or emptied) cup, i.e. other empty set, on the left side of intersection sign.

    • Thanks, Nash! I love the power set – just can’t get enough of it (maybe because I’m so jealous, I wanna be the Power Bon).

      You write, “it can also be said that empty set is common element for empty set and any other set.” I think you might have mistyped. The empty set is a common subset of both the empty set and any other set, but not a common element. This was the point of the previous article on empty sets. There is not an empty set inside the empty cup – the empty cup IS the empty set.

      I think you might mean that the empty set is a common element of the power sets of the empty set and any other set.

      Power set of empty set = { {} } (the set containing the empty set)
      Power set of {T, S, R} = { {T, S, R}, {T, S}, {T, R}, {R, S}, {T}, {R}, {S}, {} } (what you wrote out)

      In which case the empty sets are elements of the bigger sets – and are the intersection of the two.

  2. Fantastic examples and graphics! A person could literally scroll through and learn about unions, intersections, and the empty set just through the pictures and reading the captions. Super brilliant.

    My suggestion? “Write” a math picture book just with labels, captions and titles. It will be like summarizing novels in 160 character tweets! (or is it 140?)

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