To make the union of two sets, we need to put together all the elements in both sets into a single one avoiding repetitions. This is easy to explain with some images.

For example, on a picnic, my mom and my aunt brought all the food. The set M, M stands for Mom, are the different kinds of foods my mom got. Let’s make it clear; my mom does not bring a single muffin for a picnic. That would be funny, right? Nope, the muffin represents the food type in her basket, regardless of quantity. The set A (Aunt) are the kinds of foods my aunt brought.
 

Then, when they took everything out of their baskets, and it’s time to lunch, the set MA, the union of M and A, is the set of all the kinds of foods we had at the picnic.

Maybe we would expect MA to have 9 elements since M has 5 and A has 4. However, MA only has 7 elements because we should not repeat the kinds of foods that my mom and my aunt brought. That is, in MA will only appear an apple, and a bunch of bananas, representing each type of these foods.
 

If instead, we would like to know which kinds of foods in common my mom and aunt brought, then we should make the intersection of M and A. This operation is written as MA. In our example, shown in the blue shaded part of the picture above, MA consists of an apple and a bunch of bananas.

In the picture at the beginning of the lesson, we have two sets: a set of animals, and a set of flying things. We can see that a bat is at the intersection of the two sets since it is an animal that flies.

Therefore, to make the intersection of two sets, we need to choose all the elements that belong to both of them, and form a new set with those elements.

We can also encourage kids to make up sets out of some elements to turn out to be in the intersection of the two sets. For example, ask them to think of two sets: tomatoes and strawberries are in their intersection. Then, they would come up with different ideas, such as intersecting the set of red things with the set of food; or, as in the image below, they may decide that tomatoes and strawberries belong to the intersection of the set of food and the set of plants.
 

An easy and fun way of explaining these two operations to kids is through games with toys, cards, crayons, among others. For example, one can make two sets of crayons, sets A and B, and determine the different colors in each set. Then, the kids may form a new set by putting all the different colors from both sets, without repetitions, into a single one representing their union. Also, they may determine the colors that appear both in A and B to make a new set of crayons representing their intersection.

We can take a very handy approach with kids to show them the difference between union and intersection using wire hangers. Take, for example, two wire hangers and shape them as a square and as a circle to distinguish them. Then, put some buttons inside each of them, letting some in the intersection, as in the picture below.
 

Then, we can ask kids several questions starting from this. For example, how many green buttons are there in the square hanger? If their answer is 2, then one can slowly pick up the circle-shaped hanger, and they can see clearly that there are 4. We can also ask them how many red and green buttons are all together. If the answer is six green and five red buttons, then the concept of union must be pretty clear at this point! As for intersection, we may ask, for example, how many red buttons belong to both circle and square?, expecting to have “one” as an answer.

By playing with cards, we can translate these operations into a more mathematical frame.
 

For example, A in the figure above represents the set of natural numbers {4, 8, 9}, and B represents the set {2, 3, 5, 8}. Hence, AB is {2, 3, 4, 5, 8, 9} where we put the elements of A and the elements of B (without repetitions). Further, AB is {8} which contains the only element that belongs to both sets.

We can go further and make the union and intersection of three or more sets. To make the union of more than two sets, we proceed as before, forming a new set with every element of each set and avoiding repetition of the elements. As for the intersection, we should form a new set with each element that is common in all of the sets. For example, consider A and B as before, and let C be the set {2, 4, 8, 10}. Then, the union ABC of the three sets A, B and C, is the set {2, 3, 4, 5, 8, 9, 10}, where we add the new element 10 from C to AB. As for the intersection ABC of A, B and C, we see that 8 is the only element that is simultaneously in the three sets. Thus, ABC is again {8}.

Although we have finished making the intersection of the three sets, we can take this opportunity to make all the pairwise intersections and unions between A, B and C that we did not do before. That is, AC is {2,4,8,9,10}, BC is {2,3,4,5,8,10}, AC is {4,8}, and BC is {2,8}.

As we did for two sets, we can represent the intersection of three sets with diagrams. But then, we would like to represent all possible intersections between the three sets. We should be very careful to do so! First, we need to look for common elements in every two sets and those that belong to more than two sets. For example, in the image below, we have three sets containing the colors in the flags of the United States (A), Germany (G), and Brazil (B), respectively. We can see that GB only contains the color yellow, AG only includes the color red, while colors white and blue are both in AB. We should also verify if there is a color belonging to each of the three sets, if there is an element in AGB. As the picture shows, in this case, AGB has no elements because there is not a color simultaneously in the three flags of these countries.

Parents or older siblings can play this game with kids at home. Take, for example, paper plates, cut a circular frame out of them, and write the name of a country in each frame. Use tissue paper balls, paperclips, pom pom balls, or any other small craft elements of different colors to represent the colors in each flag. Then, randomly pick up three different frames, and start putting the colors inside the corresponding country, taking into account all possible intersections. If you do not know the colors on the flags by heart, you will also need a picture with the flags to play this game!

Below there is a choice of countries where all possible intersections turn out to have elements. Notice that if M stands for Mexico, then AM and BM have two elements each, and AMB consists of the color white.

We can continue making these operations with four and more sets once kids perfectly understand the intersection between two and three sets. Just remember, if we want to represent the intersection of four sets, for example, we should consider the pairwise intersections, the intersections of every three different sets (as explained above), and then the intersection of the four sets. In the following example, we explain one such representation with a game that can also be played at home, writing words on pieces of paper.

Let A be the set of words that start with “a”, C the set of words starting with “c”, R the set of words that end with “r”, and I the set of words that have at least an “i” on them. Since no word can start with “a” and “c”, none of the intersections AC, ACR, ACI, and ACRI have elements. All other possible intersections will have plenty of elements, and it is a great exercise, both for English and math, to ask children to find words that belong to them. We will put some in the following image for inspiration.

Finally, an easy way of distinguishing the symbol of intersection from the one of union, is remembering that the symbol used for union is a big “U”, and the one for intersection is like a small “N” that kids can associate with “narrow tunnel” where only one unique element can fit through.

I hope this article helps. Below is a link to interactive practices for union and intersection of sets, or use worksheets below to have your students learn the old-fashioned way, which is proved to be working.