S. D. V. Stephens

notes / Integrated Mathematics / chapter 7: Sets and Proof

7.1 Sets: Notation, Subsets and Operations

We begin with an introduction to sets. (Despite conventional wisdom, I will not start with the Peano axioms.)

Sets and their elements

Definition 7.1.1. A set is a collection of objects, its elements (or members).

Sets can be of names, dogs, cats, numbers, curves, geometric figures, and more. When nothing special is assumed about a set, it is known as an abstract set. Abstract set theory is a subfield of set theory.

Example 7.1.2. , , .

Four pictures of sets as circles: one holding a cat, a fork, a watermelon and a chair; one labelled all numbers; two overlapping circles; and two separate circles that do not touch.

Notation

Sets are commonly denoted by capital letters: . Elements are commonly denoted by lower-case letters: .

  • “” means “ belongs to ”.
  • “” means “ does not belong to ”.

Example 7.1.3. The set of all positive even integers less than : The set of all positive even integers: where “” means “and so on”.

Listing out the elements/members of a set within braces is referred to as roster notation.

when all elements are the same, and when the sets are unequal (non-identical).

Example 7.1.4. and , thus .

and , thus .

Subsets

The set is a subset of . We write being a subset of as .

Note that and can hold simultaneously. In such a case we would say that if and , then .

If but , this is known as a proper subset and is written as .

Set-builder notation

The main or fixed set we are concerned with at any given time is known as the universal set. means “all of the elements that are in and satisfy condition .”

Sometimes the universal set is understood by implication, and then is sufficient. Additionally, the set of all positive real numbers can be written as , where we assume . Reminder that is a placeholder, and thus , etc.

Special sets

Note. The cardinality or size of a set is written . For example, if , then ; and if , then .

The empty set

At times, a set will contain no elements. This set is a special set known as the empty set, denoted by . is a subset of every set.

If a set is a box, then the empty set is an empty box. Side note: is a Norwegian symbol and was made the notation for the empty set by the secret society known as “Nicolas Bourbaki.”

Note that , as a hat is not the same as a box with a hat in it. Similarly, : has no elements, while has one element (the empty set). (Damn, I really ate with that explanation.)

Unions, intersections and differences

Four diagrams of two sets A and B drawn as circles: overlapping circles with both shaded for the union; overlapping circles with only the overlap shaded for the intersection; two separate circles for disjoint sets, whose intersection is empty; and two separate shaded circles showing that a union can also look like this.
  • : the union.
  • : the intersection.
  • : and are disjoint.

If and , then the union is , or .

The set of elements of that are not in is

Unions and intersections are similar to addition and multiplication in that they are both commutative and associative:

Classes of sets

A collection of sets is known as a class. Cursive capital letters are used to denote classes.

The union of all the sets in the class (meaning , or ) is written the second form only if is finite and .

(Okay, what if we said ??)

Remark (Correction). The class (with the irrationals) has three members, each a set of numbers, so it does not equal . What is true is that the union of its members is : , since .

The exact same type of “summation” works for intersections:

Exercises

Exercise 7.1.5. Use roster notation for the sets

Show solution
  • : , so .
  • : , so .
  • : , so .
  • : . Either , or , i.e. . So .
  • : , so .
  • : . The quadratic formula on gives , i.e. and . Then , , so and , and . (F, circled.)

Remark (Correction). The set is wrong. From we get or . The first gives , so or ; the second gives . So , and none of lies in (for example ).

Exercise 7.1.6. List the inclusion relations among the sets of the previous exercise.

Show solution

, , , , .

Remark (Correction). With the corrected from the previous exercise, three more inclusions hold: , and .

Exercise 7.1.7. Let , . True or false?

Show solution
Statement Reason
(a) , and ✓
(b) ✓
(c) ? ✗
(d) ✓
(e) is a set? No. ✗
(f) is not a set. ✗

Exercise 7.1.8. Let , . True or false?

Show solution
Statement Reason
(a) and ✓
(b) ✓
(c) ✓
(d) ✓
(e) ✗
(f) ✗

Remark (Correction). The answers are right, but the reasons given for (a) and (b) belong to (c). For (b), every element of lies in : the only element of is , and . For (a), holds and because while . The fact says that itself is an element of , which is the reason for (c).

Exercise 7.1.9. List all subsets of .

Show solution

Remark (Correction). This list repeats itself and includes sets that are not subsets of : item 9 is again (item 6), items 10 to 13 have as an element and so are not subsets of , and items 14 to 16 each name two sets at once. The subsets of are ; ; ; ; and .

Exercise 7.1.10. Let , , , . True or false?

Show solution
Statement Reason
(a) ? ✗
(b) ? ✗
(c) ? ✗
(d) ✓
(e) , and ✗
(f) , and ✓

Remark (Correction). The ✗ in (e) is correct, but the reason shown writes as , which is . In fact , and would require the numbers and to be elements of , which they are not. What is true is .

Exercise 7.1.11. Prove that (a) , (b) , (c) if and only if .

Show solution

(a) : any set is equal to the set .

(b) : associativity/commutativity of sets.

(c) if : equivalence.

Exercise 7.1.12. Commutative laws: , .

Exercise 7.1.13. Associative laws: , .

Exercise 7.1.14. Distributive laws: , .

Exercise 7.1.15. If is a class of sets, then