Sets

What are Sets?

Mathematically, a set is a collection of mathematical objects, which are called elements of the set. Those elements can be anything, but are often numbers, geometric objects, functions, or other sets.

Why are sets important to mathematicians?

• Set theory forms the basis of all of mathematics.
• Set theory is used to discuss infinity.
• Set notation is used commonly in mathematical proofs of all kinds.

Why is it important to teach about sets?

• The foundations of mathematics is built on set theory.
• The ideas of sets and set notation come up frequently in multiple areas of science and technology.
• Sets are used to discuss the relationships between different types of numbers, such as natural numbers, irrational numbers, real numbers and complex numbers.
• Set notation is commonly used in statements of proofs.

What is important to teach about sets?

Students should understand the definitions of:

• set
• element
• subset
• member
• union
• intersection
• complement
• superset
• finite set
• infinite set
• empty set

Students should also have an understanding of:

• set notation
• Venn diagrams
• equality of sets
• cardinality of sets

Examples of ways sets will be used in high school STEM classes:

• A circle can be defined as the set of all points that are the same distance away from a point.
• Natural numbers, whole numbers, rational numbers, irrational numbers, real numbers and complex numbers are all sets of numbers that will be encountered and their relationships will be discussed in terms of sets and written in set notation.
• Sets are used frequently in computer programming.
• Functions can be defined as a way to map the elements of one set onto another set.
• Even if the terminology is different, the logical thinking involved in manipulating sets is often used in talking about many scientific concepts.
• Relationships between many scientific concepts are illustrated with Venn diagrams.

Teaching Hints:

• The general idea of a set, union, intersection, subset, etc., are all pretty intuitive. If a student is struggling, it may be due to the introduction of new words or new symbols along with the concepts. It may be helpful to introduce the concepts using more common language, before focusing on the terms. Here are some words to substitute:
  • “group” for “set”,
  • “belongs to” for “element”
  • “in both” for “intersection”
  • “in either” for “union”
  • “smaller group within the other group” for “subset”
  • “everything not in the set” for “complement”
  • “size” for “cardinality”
• But, students must become fluent in using these terms and especially the notation. The notation will be used frequently throughout future math lessons and fluency will greatly impact how quickly new concepts are grasped.

More advanced topics:

Some students will benefit from at least being aware of the following:

• De Morgan’s laws
• Partitions
• Power sets

Non-transferable skills:

Concepts that may be interesting to some students but are generally more esoteric and not necessary for a solid understanding of STEM include:

• Axiomatic set theory, upon which the foundations of math are built.
• The idea that some infinite sets are bigger than others.
• The continuum hypothesis, which is that there is no set bigger than the natural numbers but smaller than the real numbers.
• Russell’s paradox
• Cantor’s paradox

Subtopics

• Set notation
• Set operations
• Classification of numbers

Related topics

• Infinity