Construct a Venn diagram illustrating the sets below. U = {1, 2, 3, 4, 5, 6, 7, 8, 9}, W = {2, 4, 5, 6, 8}, X = {1, 2, 5, 7, 8}, Y = {2, 3, 4, 7, 8}

Steps to Solve

1. Identify the universal set and individual sets

The universal set $U$ is ${1, 2, 3, 4, 5, 6, 7, 8, 9}$. The individual sets are $W = {2, 4, 5, 6, 8}$, $X = {1, 2, 5, 7, 8}$, and $Y = {2, 3, 4, 7, 8}$.

2. Find the intersections between each pair of sets

$W \cap X = {2, 5, 8}$ $W \cap Y = {2, 4, 8}$ $X \cap Y = {2, 7, 8}$

3. Find the intersection of all three sets

$W \cap X \cap Y = {2, 8}$

4. Determine elements unique to each set

Elements only in $W$: $W - (W \cap X) - (W \cap Y) + (W \cap X \cap Y) = {6}$ Elements only in $X$: $X - (W \cap X) - (X \cap Y) + (W \cap X \cap Y) = {1}$ Elements only in $Y$: $Y - (W \cap Y) - (X \cap Y) + (W \cap X \cap Y) = {3}$

5. Determine elements in exactly two sets but not the third

Elements in $W$ and $X$ but not $Y$: $(W \cap X) - (W \cap X \cap Y) = {5}$ Elements in $W$ and $Y$ but not $X$: $(W \cap Y) - (W \cap X \cap Y) = {4}$ Elements in $X$ and $Y$ but not $W$: $(X \cap Y) - (W \cap X \cap Y) = {7}$

6. Determine elements in the universal set but not in any of the three sets

$U - (W \cup X \cup Y) = {9}$

7. Populate the Venn diagram

Place the elements in the appropriate regions of the Venn diagram based on the intersections and unique elements determined in the previous steps:

• $W$ only: ${6}$
• $X$ only: ${1}$
• $Y$ only: ${3}$
• $W \cap X$ only: ${5}$
• $W \cap Y$ only: ${4}$
• $X \cap Y$ only: ${7}$
• $W \cap X \cap Y$: ${2, 8}$
• $U - (W \cup X \cup Y)$: ${9}$