From the Venn-diagram given alongside answer the following questions: a) What is the cardinal number of the universal set? Write it.

Here are the solutions to the questions based on the provided Venn diagram.

The elements in the Venn diagram are: Universal Set $U = \{a, b, c, d, e, f, g, i, m\}$ Set $G = \{a, b, c, g, i\}$ Set $H = \{c, d, e, g\}$ Set $K = \{f, g, i\}$

a) What is the cardinal number of the universal set U?

Step 1: List all distinct elements in the universal set U from the Venn diagram. $U = \{a, b, c, d, e, f, g, i, m\}$

Step 2: Count the number of elements in U. $|U| = 9$ The cardinal number of the universal set U is $\boxed{*9*}$.

b) Write the set $(G \cap K)$ by listing method.

Step 1: Identify the elements in set G. $G = \{a, b, c, g, i\}$

Step 2: Identify the elements in set K. $K = \{f, g, i\}$

Step 3: Find the intersection of G and K, which are the elements common to both sets. $G \cap K = \{g, i\}$ The set $(G \cap K)$ is $\boxed{*\{g, i\}*}$.

c) Prove that: $\overline{G \cup H} = \overline{G} \cap \overline{H}$

Step 1: Find $G \cup H$. $G \cup H = \{a, b, c, g, i\} \cup \{c, d, e, g\}$ $G \cup H = \{a, b, c, d, e, g, i\}$

Step 2: Find the complement of $G \cup H$, denoted as $\overline{G \cup H}$. $\overline{G \cup H} = U - (G \cup H)$ $\overline{G \cup H} = \{a, b, c, d, e, f, g, i, m\} - \{a, b, c, d, e, g, i\}$ $\overline{G \cup H} = \{f, m\}$

Step 3: Find the complement of G, denoted as $\overline{G}$. $\overline{G} = U - G$ $\overline{G} = \{a, b, c, d, e, f, g, i, m\} - \{a, b, c, g, i\}$ $\overline{G} = \{d, e, f, m\}$

Step 4: Find the complement of H, denoted as $\overline{H}$. $\overline{H} = U - H$ $\overline{H} = \{a, b, c, d, e, f, g, i, m\} - \{c, d, e, g\}$ $\overline{H} = \{a, b, f, i, m\}$

Step 5: Find the intersection of $\overline{G}$ and $\overline{H}$, denoted as $\overline{G} \cap \overline{H}$. $\overline{G} \cap \overline{H} = \{d, e, f, m\} \cap \{a, b, f, i, m\}$ $\overline{G} \cap \overline{H} = \{f, m\}$

Step 6: Compare the results from Step 2 and Step 5. Since $\overline{G \cup H} = \{f, m\}$ and $\overline{G} \cap \overline{H} = \{f, m\}$, Therefore, $\boxed{\overline{G \cup H} = \overline{G} \cap \overline{H}}$ is proven.

d) What percent of the elements of universal set U have $(G \cup H \cup K)$?

Step 1: Find the union of sets G, H, and K. $G \cup H \cup K = \{a, b, c, g, i\} \cup \{c, d, e, g\} \cup \{f, g, i\}$ $G \cup H \cup K = \{a, b, c, d, e, f, g, i\}$

Step 2: Find the cardinal number of $G \cup H \cup K$. $|G \cup H \cup K| = 8$

Step 3: Recall the cardinal number of the universal set U. $|U| = 9$

Step 4: Calculate the percentage of elements in $U$ that are in $(G \cup H \cup K)$. $\text{Percentage} = \frac{|G \cup H \cup K|}{|U|} \times 100\%$ $\text{Percentage} = \frac{8}{9} \times 100\%$ $\text{Percentage} \approx 0.8889 \times 100\%$ $\text{Percentage} \approx 88.89\%$ The percentage of elements of universal set U that have $(G \cup H \cup K)$ is approximately $\boxed{*88.89\%*}$.