Three ways two sets can be related A Venn diagram shows how two sets are related. Each relationship is a statement about the overlap, so read the statement first and then decide the picture. **Statement in words** | **Set notation** | **Picture** No element is in both sets. | A B= | Disjoint: two separate circles. Every element of B is also in A. | B A, so A B=B | Subset: B drawn inside A. Some elements are in both sets, and each set has an element outside the other. | A B!=, A B, B A | Overlapping circles. Disjoint: A B= Subset: B A Overlapping: A B!= Why the picture changes: each condition says that one region of the overlapping diagram is empty. A B= empties the overlap, so the circles are drawn apart. B A empties the part of B outside A, so B is drawn inside A. When no region is forced to be empty, the circles overlap. If A=B, one circle carries both labels. Method - Find A B, by listing or from the given statement. If it is empty, draw the sets apart. - If it is not empty, check whether every element of one set belongs to the other. If so, draw the smaller set inside the larger one. - Otherwise draw overlapping circles. - With a third set, decide its relationship with each of the other sets in turn, then draw it. Example xi=x: x is an integer, 1 <= x <= 10, A=2,4,6,8,10, B=4,8 and C=1,3,5. Draw a Venn diagram to show the relationships between A, B and C. Every element of B is in A, so B A: draw B inside A. A C=, so C is drawn apart from A, and then B C= as well. Check Read every condition back from your diagram: an overlap appears only where the sets share an element. Sets with different numbers of elements can still overlap or be nested; only a common element, or the lack of one, decides.

Representing relationships between sets

Secondary 4

Three ways two sets can be related

A Venn diagram shows how two sets are related. Each relationship is a statement about the overlap, so read the statement first and then decide the picture.

Statement in words	Set notation	Picture
No element is in both sets.	$A\cap B=\varnothing$	Disjoint: two separate circles.
Every element of $B$ is also in $A$.	$B\subseteq A$, so $A\cap B=B$	Subset: $B$ drawn inside $A$.
Some elements are in both sets, and each set has an element outside the other.	$A\cap B\neq\varnothing$, $A\not\subseteq B$, $B\not\subseteq A$	Overlapping circles.

Disjoint: $A\cap B=\varnothing$

Subset: $B\subseteq A$

Overlapping: $A\cap B\neq\varnothing$

Why the picture changes: each condition says that one region of the overlapping diagram is empty. $A\cap B=\varnothing$ empties the overlap, so the circles are drawn apart. $B\subseteq A$ empties the part of $B$ outside $A$, so $B$ is drawn inside $A$. When no region is forced to be empty, the circles overlap. If $A=B$, one circle carries both labels.

Method

Find $A\cap B$, by listing or from the given statement. If it is empty, draw the sets apart.
If it is not empty, check whether every element of one set belongs to the other. If so, draw the smaller set inside the larger one.
Otherwise draw overlapping circles.
With a third set, decide its relationship with each of the other sets in turn, then draw it.

Example

$\xi=\{x: x \text{ is an integer}, 1\le x\le 10\}$, $A=\{2,4,6,8,10\}$, $B=\{4,8\}$ and $C=\{1,3,5\}$. Draw a Venn diagram to show the relationships between $A$, $B$ and $C$.

Every element of $B$ is in $A$, so $B\subset A$: draw $B$ inside $A$. $A\cap C=\varnothing$, so $C$ is drawn apart from $A$, and then $B\cap C=\varnothing$ as well.

Check

Read every condition back from your diagram: an overlap appears only where the sets share an element. Sets with different numbers of elements can still overlap or be nested; only a common element, or the lack of one, decides.

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Statement in words	Set notation	Picture
No element is in both sets.	\(A\cap B=\varnothing\)	Disjoint: two separate circles.
Every element of \(B\) is also in \(A\).	\(B\subseteq A\), so \(A\cap B=B\)	Subset: \(B\) drawn inside \(A\).
Some elements are in both sets, and each set has an element outside the other.	\(A\cap B\neq\varnothing\), \(A\not\subseteq B\), \(B\not\subseteq A\)	Overlapping circles.