The game SET® is an excellent way to introduce basic set theory. It provides a concrete model for understanding and a tool for working through set operations. Students should be encouraged to use the cards when trying to complete the exercises.

D = the set of all the cards in the deck of the game SET®
R = the set of red cards
G = the set of green cards
P = the set of purple cards
1 = the set of cards with one shape
2 = the set of cards with two shapes
3 = the set of cards with three shapes
o = the set of cards with ovals
~ = the set of cards with squiggles
= the set of cards with diamonds
L = the set of cards with light shading
M = the set of cards with medium shading
H = the set of cards with heavy shading

Cardinality
A set, in general, is any collection of objects. One of the most basic ways we have of describing sets is cardinality. Cardinality is simply the number of elements or objects in a set. Another name for the cardinality of a set is the set's cardinal number. We use the symbol |X| to mean the cardinality of a set or cardinal number of some set X. For example, |R| means the number of objects (or cards) in the set R, which we have defined as the set of cards that are red in the game SET®.

|R| = 27
|M| = 27
|1| = 27

Union:
Union is a set operation, or a way of relating two sets together. The easiest way to think of union is that for any two sets, their union includes all of the elements that are in one or both of the sets. The symbol for union is . When you think of union, you should think of the word "or". For example (R ~) means all the cards that are red or squiggles or both.

We can also use cardinality with union. For example, | R ~ | = 45, because there are 27 red cards and 27 cards with squiggles which adds up to 54, but since there are 9 cards with red squiggles that are counted twice, we subtract the number of red squiggles (9). In other words, we add the number of elements in each set and then subtract the number of elements that the two sets have in common. The empty set: = the empty set; a set with no elements.

Exercises:
For each exercise:

Example:
G
a. the set of cards that are green or are diamond
b. (27 green cards) + (27 diamonds) - (9 green diamonds) |G | = 45

1. R ~

2. M 1

3. 2 o

4. P H

5. o P

6. L ~

7. R 3

8. P G R

9. 1 2 3

10. ~ o

Intersection:
Intersection is another set operation or way of relating two sets together. The intersection of two sets is the elements that are in both sets, or the elements the two sets have in common. The symbols for intersection is . When you think of intersection you should think of the word "and". For example, (G 1) means the set of cards that are green and have 1 shape.

We can use cardinality with intersection. Let's say that we wanted to know how many cards are green and have one shape (G 1). We could find each of the cards and count them or we could use what we know about the game SET®. We know that there are three different shapes and for each shape there are three different shadings. Whether we count the cards or try to "think out" the problem, we come up with 9 cards.

Exercises
For each exercise:

a. write in words what the symbols mean, and
b. give the cardinal number

11. R ~

12. 2

13, G P

14. H ~

15. o R

16. P 3

17. R 1 o

18. G 2

19. (R 1) ~ [hint: do what is in the parentheses first]

20. P (2 o)

 

Symmetric Difference
Symmetric difference is another set operation. The simplest way to think of symmetric difference is that it is all the elements that are in either one set or the other but not in both. The symbol for symmetric difference is . Take the example (R ~). In words this means all the cards with red shapes or all the cards with squiggles, but not the cards with red squiggles. To find the cardinal number for (R ~) simply add the number of R (red cards) to the number of ~ (squiggles) then subtract the number of red squiggles. There are 27 red cards and 27 squiggles which adds up to 54. There are 9 red squiggles and since the red squiggles are in both the set of red cards and the set of squiggles, we must subtract them twice (54-18). Therefore the number of elements is 36.

Exercises
For each exercise:
a. Write in words what the symbols mean, and
b. give the cardinal number

Example
H R
a. The set of cards that have heavy shading and cards that have one shape, but not the cards that have heavy shaded one shapes.
b. (27 heavy shaded cards) + (27 red cards) - (9 heavy shaded red cards from the set of reds) - (9 heavy shaded red cards from the set of heavy shaded cards) = 36 | H R | = 36 21. P G

22. D o

23. 1 3

24. L 1

25. ~ 2

26. H M

27. R ~

28. (P R) G [hint: remember to treat (P R) as one set]

29. (R G) 2

30. (~ P)

Complement
The complement of a particular set is simply all the elements in the universal set that are not in that set. When we are using the game SET®, the universal set is the whole deck of cards. Take the set P (purple cards). The complement of P (P') is all the cards that are not purple or, in other words, all the cards that are red or green. The cardinal number of P' (|P'|) is the number of elements in the universe (D) minus the number of elements in P.

|D| - |P| = |P'|
81 - 27 = 54

Exercises
For each exercise:
a. write in words what the symbols mean, and
b. give the cardinal number

Example
(R ~)'
a. all the cards that are not red squiggles
b. |D| - |(R ~ )| = (R ~)'
81 - 9 = 72

31. ~'

32. 2'

33. H'

34. (H 1)'

35. (P )'

36. D'

37. (G o)'

38. (R L)'

39. (R G P)'

40. (R 1)'