IB MYP 3 Mathematics 2.3 Venn Diagrams and Regions Study Notes - New Syllabus

2.3 – Venn Diagrams and Regions

A Venn diagram is a diagram used to represent sets and show the relationships between them. It makes it easier to identify elements that belong to one set, both sets, neither set, or a particular region.

 Parts of a Venn Diagram

 One-Set Venn Diagram

A single set \(A\) is represented by a circle inside the universal set \(U\).

The elements inside the circle belong to \(A\), while elements outside the circle but inside the rectangle belong to \(A’\).

 Two-Set Venn Diagrams

Two sets \(A\) and \(B\) are represented by two circles inside the universal set.

There are four important regions:

Overlapping Sets

When two sets have elements in common, their circles overlap.

The overlapping region represents:

\(A\cap B\)

For example, suppose:

\(A=\{1,2,3,4,5\}\)

\(B=\{4,5,6,7\}\)

Then:

\(A\cap B=\{4,5\}\)

Therefore, \(4\) and \(5\) must be placed in the overlapping region.

Subset Venn Diagram

If every element of \(A\) is also an element of \(B\), then \(A\) is a subset of \(B\):

\(A\subseteq B\)

In a Venn diagram, \(A\) is drawn completely inside \(B\).

For example:

\(A=\{2,4\}\)

\(B=\{2,4,6,8\}\)

Since every element of \(A\) is in \(B\):

\(A\subseteq B\)

In this case, there is no separate \(A\)-only region because everything in \(A\) is also in \(B\).

Disjoint Sets

If two sets have no elements in common, they are disjoint.

Their circles do not overlap.

\(A\cap B=\varnothing\)

For example:

\(A=\{1,3,5,7\}\)

\(B=\{2,4,6,8\}\)

No element belongs to both sets, so they are disjoint.

Shading Regions in Venn Diagrams

Venn-diagram questions may ask you to shade a particular set or combination of sets.

 Placing Numbers in the Correct Regions

When a question gives a list of numbers and asks you to place them in a Venn diagram, check each number against the definitions of the sets.

For example, let \(U\) contain the whole numbers from \(1\) to \(12\).

Let:

\(A=\text{even numbers}\)

\(B=\text{multiples of }3\)

A number belongs in the overlap if it is both even and a multiple of \(3\).

The numbers \(6\) and \(12\) satisfy both conditions, so:

\(A\cap B=\{6,12\}\)

The remaining even numbers go in the \(A\)-only region, while the remaining multiples of \(3\) go in the \(B\)-only region.

Number of Elements in Each Region

For two sets, the total number of elements can be divided into four regions:

If \(n(A)\), \(n(B)\), and \(n(A\cap B)\) are known:

\(\text{A only}=n(A)-n(A\cap B)\)

\(\text{B only}=n(B)-n(A\cap B)\)

And:

\(\text{Neither}=n(U)-n(A\cup B)\)

This allows a Venn diagram to be completed even when some regions are initially unknown.