Find the union of the sets {5, 6, 8, 9} and {5, 7, 8}.

Step-by-step solution

1. 1

   Recall the definition. The union $A\cup B$ contains every element that is in $A$, or in $B$, or in both. It is an inclusive or, so nothing gets thrown away — the union is always at least as big as either set.

2. 2

   Sweep the first set. From $\{5,6,8,9\}$ take $5$, $6$, $8$ and $9$. The running collection is

   $\{5,6,8,9\}$

3. 3

   Sweep the second set, skipping repeats. From $\{5,7,8\}$: $5$ is already listed, $7$ is new, $8$ is already listed. A set records only whether an element belongs, never how many times, so duplicates are dropped:

   $\{5,6,8,9\}\cup\{7\}=\{5,6,7,8,9\}$

4. 4

   Write the answer in increasing order. Order does not change a set, but listing elements from small to large makes an omission easy to spot:

   $\{5,6,8,9\}\cup\{5,7,8\}=\{5,6,7,8,9\}$

5. 5

   Check the count. The two sets have $4$ and $3$ elements, sharing the two elements $5$ and $8$. Inclusion-exclusion predicts

   $|A\cup B|=4+3-2=5$

   and the answer indeed has $5$ elements $\checkmark$. The union is certainly not the empty set — a common slip when the two sets overlap heavily — so the box holds $\{5,6,7,8,9\}$.