Four regions of a two-set Venn diagram Two overlapping sets A and B split the universal set xi into four regions that do not overlap. Every element of xi lies in exactly one of them, so every two-set expression is just a choice of regions. **Region** | **Inside or outside A?** | **Inside or outside B?** | **Set notation** R_1 | inside | outside | A B' R_2 | inside | inside | A B R_3 | outside | inside | A' B R_4 | outside | outside | (A B)' So A is R_1 and R_2, B is R_2 and R_3, A' is R_3 and R_4, and B' is R_1 and R_4. When one set lies inside the other, or the sets share no element, a region is empty and is not drawn. The remaining regions are read in exactly the same way. B A: there is no R_3, so A' B=. A B=: there is no R_2. Shading an expression: the tick method - Write down the regions of each set or complement in the expression. - Tick those regions in a table, one column for each part of the expression. - For (or), shade every region with at least one tick. For (and), shade only the regions ticked in both columns. - For a complement of a bracket, such as (A B)', find the regions of the bracket first, then shade every other region. Example Shade A' B'. A' is R_3 and R_4; B' is R_1 and R_4. The expression uses, so shade the region ticked twice. **Region** | **A'** | **B'** | **Shade?** R_1 | – | | no R_2 | – | – | no R_3 | | – | no R_4 | | | yes Only R_4 is shaded, so A' B'=(A B)': outside A and outside B is the same as outside the union. Describing a shaded region - For each shaded region ask: inside or outside A? Inside or outside B? Write A or A', and B or B', joined by. - If more than one region is shaded, join the descriptions with, then look for a shorter name. - If every region except one is shaded, describe the unshaded region and take its complement. Example Describe each shaded region in set notation. Left: R_1 and R_3 are shaded. Right: every region except R_2 is shaded. Left: R_1 is inside A and outside B, A B'; R_3 is A' B. The region is (A B') (A' B), the elements in exactly one of the two sets. Right: only R_2, which is A B, is unshaded, so the shaded region is (A B)'. Check Test one region against your answer before you finish. Brackets change the region: (A B)' is R_1, R_3 and R_4, but A' B' is R_4 only. Shade only inside the rectangle xi, and on a nested or disjoint diagram skip any region that is not drawn.

Shading and describing Venn regions

Secondary 4

Four regions of a two-set Venn diagram

Two overlapping sets $A$ and $B$ split the universal set $\xi$ into four regions that do not overlap. Every element of $\xi$ lies in exactly one of them, so every two-set expression is just a choice of regions.

Region	Inside or outside $A$?	Inside or outside $B$?	Set notation
$R_1$	inside	outside	$A\cap B'$
$R_2$	inside	inside	$A\cap B$
$R_3$	outside	inside	$A'\cap B$
$R_4$	outside	outside	$(A\cup B)'$

So $A$ is $R_1$ and $R_2$, $B$ is $R_2$ and $R_3$, $A'$ is $R_3$ and $R_4$, and $B'$ is $R_1$ and $R_4$.

When one set lies inside the other, or the sets share no element, a region is empty and is not drawn. The remaining regions are read in exactly the same way.

$B\subseteq A$: there is no $R_3$, so $A'\cap B=\varnothing$.

$A\cap B=\varnothing$: there is no $R_2$.

Shading an expression: the tick method

Write down the regions of each set or complement in the expression.
Tick those regions in a table, one column for each part of the expression.
For $\cup$ (or), shade every region with at least one tick. For $\cap$ (and), shade only the regions ticked in both columns.
For a complement of a bracket, such as $(A\cup B)'$, find the regions of the bracket first, then shade every other region.

Example

Shade $A'\cap B'$. $A'$ is $R_3$ and $R_4$; $B'$ is $R_1$ and $R_4$. The expression uses $\cap$, so shade the region ticked twice.

Region	$A'$	$B'$	Shade?
$R_1$	–	$\checkmark$	no
$R_2$	–	–	no
$R_3$	$\checkmark$	–	no
$R_4$	$\checkmark$	$\checkmark$	yes

Only $R_4$ is shaded, so $A'\cap B'=(A\cup B)'$: outside $A$ and outside $B$ is the same as outside the union.

Describing a shaded region

For each shaded region ask: inside or outside $A$? Inside or outside $B$? Write $A$ or $A'$, and $B$ or $B'$, joined by $\cap$.
If more than one region is shaded, join the descriptions with $\cup$, then look for a shorter name.
If every region except one is shaded, describe the unshaded region and take its complement.

Example

Describe each shaded region in set notation.

Left: $R_1$ and $R_3$ are shaded.

Right: every region except $R_2$ is shaded.

Left: $R_1$ is inside $A$ and outside $B$, $A\cap B'$; $R_3$ is $A'\cap B$. The region is $(A\cap B')\cup(A'\cap B)$, the elements in exactly one of the two sets.

Right: only $R_2$, which is $A\cap B$, is unshaded, so the shaded region is $(A\cap B)'$.

Check

Test one region against your answer before you finish. Brackets change the region: $(A\cap B)'$ is $R_1$, $R_3$ and $R_4$, but $A'\cap B'$ is $R_4$ only. Shade only inside the rectangle $\xi$, and on a nested or disjoint diagram skip any region that is not drawn.

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Region	Inside or outside \(A\)?	Inside or outside \(B\)?	Set notation
\(R_1\)	inside	outside	\(A\cap B'\)
\(R_2\)	inside	inside	\(A\cap B\)
\(R_3\)	outside	inside	\(A'\cap B\)
\(R_4\)	outside	outside	\((A\cup B)'\)

Region	\(A'\)	\(B'\)	Shade?
\(R_1\)	–	\(\checkmark\)	no
\(R_2\)	–	–	no
\(R_3\)	\(\checkmark\)	–	no
\(R_4\)	\(\checkmark\)	\(\checkmark\)	yes