To find $|D \cup E|$, the cardinality (size) of the union of sets $D$ and $E$, we first need to determine the elements in $D \cup E$, which represents all unique elements from both sets $D$ and $E$.

Given:

• $D = \{ q, u, o, t, e \}$
• $E = \{ b, a, r, o, q, u, e \}$

Step 1: Find the Union of $D$ and $E$

The union $D \cup E$ includes all elements that are in either set $D$ or $E$ (or both), without repeating any elements.

• Elements in $D$: $\{ q, u, o, t, e \}$
• Elements in $E$: $\{ b, a, r, o, q, u, e \}$

Now combine these two sets, eliminating duplicates: $D \cup E = \{ q, u, o, t, e, b, a, r \}$

Step 2: Find the Cardinality of $D \cup E$

Now, count the unique elements in $D \cup E$: $|D \cup E| = 8$

Thus, the cardinality of $D \cup E$ is $\boxed{8}$.

Would you like any further explanation or details on this?

Here are 5 related questions for further practice:

1. What is $|D \cap E|$, the cardinality of the intersection of $D$ and $E$?
2. Find $|E^c|$, the complement of set $E$ relative to $U$.
3. How many elements are in $D^c \cup E$, where $D^c$ is the complement of set $D$?
4. If $F = \{ p, s, o, l, i, d \}$, what is $|D \cup F|$?
5. Find the symmetric difference $D \Delta E$, which is the set of elements in either $D$ or $E$, but not in both.

Tip: When combining sets, always ensure to remove duplicates to accurately determine the union.