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The Venn diagram was introduced by John Venn. Yes, the Venn diagram is named after a real person! His idea was to show sets in terms of pictures. The Venn diagram is now used in many fields, including mathematics. Let’s take a look at John Venn’s idea.

Set Review

Before we get into Venn diagrams, let’s review sets.

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A set is a collection of elements. An element is a collection of anything – numbers, letters, words, objects. Using math symbols, the symbol below means element.

Mathematical symbol for element
image of the symbol for element

A subset is a special, funny set.

If every element in one set is included in another set, they are called subsets. Using math symbols, the symbol below means subset.

Mathematical symbol for subset
symbol for subset

Let’s say our universe is the things, or elements, in my apartment.

So, our universal set U = {fork, spoon, knife, spatula, whisk, plate, cup, napkin, tablecloth, television, couch}.Let set K be a list of utensils in my kitchen. K = {fork, spoon, knife, spatula, whisk}.Let set T be a list of things, or elements, on my table. T = {plate, fork, knife, cup, napkin, tablecloth}.So, we can say that a fork is an element of K.

Using math symbols, we would write:

Fork is an element of K, using math symbols.
fork is an element of k

This is read, ‘Fork is an element of K,’ or, ‘Fork belongs to K.’ That makes sense because the fork belongs in the kitchen.

Set Operations

Let’s use sets K and T to review set operations.

Union

Union is when two or more sets are united, or added together.K union T = all of the elements of set K or all of the elements in set T, and that would equal {fork, spoon, knife, spatula, whisk, plate, cup, napkin, tablecloth}.The ‘or’ tells us that it is any object in set K, the kitchen, or set T, on the table.

Intersection

Intersection is when two or more sets intersect, or are common objects, in sets.K intersects T = all of the numbers in set K and set T, and that equals {fork, knife}.The ‘and’ tells us that it is in the kitchen and on the table at the same time.

Complement

Complement is when a set is subtracted or not included in the set.

K‘ = everything not in the kitchen, not in set K, and that equals {plate, cup, napkin, tablecloth, television, couch}.

Venn Diagram

A Venn diagram can show us the sets and operations nicely in picture form. Let’s look at union, intersection and complement using a Venn diagram. We’re going to continue using set K and set T. Below is what a Venn diagram showing the sets K and T looks like:

Venn diagram of sets K and T
Union

K union T = {fork, spoon, knife, spatula, whisk, plate, cup, napkin, tablecloth}.To show union in a Venn diagram, we highlight the circle K and circle T:

Highlighting sets in a Venn diagram shows union.
union highlighted in Venn diagram

Intersection

K intersects T = {fork, knife}.To show intersection in a Venn diagram, we highlight only where the two circles intersect each other in the middle of our picture:

Highlighting only the portions of the sets that cross each other in a Venn diagram shows intersection.
Complement

K‘ = {plate, cup, napkin, tablecloth, television, couch}.To show complement in a Venn diagram, we highlight everything except the circle that represents set K:

Highlighting everything outside the circle you are looking at shows complement.
Subset

Let’s pretend we have another set called B = {spatula, whisk}, elements we would need to bake. That means in our Venn diagram we need another circle inside of K.

A subset shows up as a circle within a circle in a Venn diagram.
Disjoint

Let’s look at the elements {couch, television}. How about we call them set P = {couch, television}. We can see that circle P isn’t inside of circles K or T. It also doesn’t intersect any of those sets:

Drawing a set outside of any other sets shows disjoint on a Venn diagram.
empty set.The symbol that looks like a zero with a line through it means the empty set. There isn’t any element that is in P and K at the same time, so it’s empty!This set P is called a disjoint set. It is disjoint, or has no elements that are the same in any other set, but it is still in our universal set.

Lesson Summary

K union T = {fork, spoon, knife, spatula, whisk, plate, cup, napkin, tablecloth}. Union is an object in set K (the kitchen) or set T (on the table).

Set B is also part of set K, so it must be highlighted, too.K intersects T = {fork, knife}. Intersection is any object that is in the kitchen and on the table at the same time.K‘ = {plate, cup, napkin, tablecloth, television, couch}. Complement is any element that is not in the set listed.

So, K‘ is all elements not in set K.

Learning Outcomes

As a result of watching this video, you might be able to:

  • Define sets, elements and subsets
  • Identify the mathematical symbols for element and subset
  • Explain what a Venn diagram represents
  • Describe how each of the following would appear on a Venn diagram: union, intersection, complement, subset and disjoint

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