To solve this problem, we need to find $A'$, which is the complement of set $A$ within the universal set $U$. The complement of $A$ consists of all elements in $U$ that are not in $A$.

Given:

• $A = \{ y, o, g, u, r, t \}$
• $U = \{ a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z \}$

Step 1: List the elements in $A$

$A$ contains: $y, o, g, u, r, t$.

Step 2: Identify elements in $U$ but not in $A$

We go through each element in $U$ and exclude those found in $A$:

So, the elements in $U$ but not in $A$ are: $A' = \{ a, b, c, d, e, f, h, i, j, k, l, m, n, p, q, s, v, w, x, z \}$

Final Answer

The complement $A'$ is: $A' = \{ a, b, c, d, e, f, h, i, j, k, l, m, n, p, q, s, v, w, x, z \}$

Would you like a breakdown of any specific steps?

Here are some additional questions you might find helpful:

1. What is a universal set and how does it affect finding complements?
2. How can we represent sets and their complements using Venn diagrams?
3. How would we handle finding the complement if $U$ were infinite (like all integers)?
4. What is the difference between a union and a complement?
5. How would the answer change if another element were added to set $A$?

Tip: When working with set complements, always double-check that you've excluded only the elements in the subset from the universal set.