From the Venn-diagram given alongside answer the following questions: a) What is the cardinal number of the…

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

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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 9. b) Write the set (G ∩ 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 ∩ K = g, i The set (G ∩ K) is g, i. c) Prove that: G ∪ H = G ∩ H Step 1: Find G ∪ H. G ∪ H = a, b, c, g, i ∪ c, d, e, g; G ∪ H = a, b, c, d, e, g, i Step 2: Find the complement of G ∪ H, denoted as G ∪ H. G ∪ H = U − (G ∪ H); G ∪ H = a, b, c, d, e, f, g, i, m − a, b, c, d, e, g, i; G ∪ H = f, m Step 3: Find the complement of G, denoted as G. G = U − G; G = a, b, c, d, e, f, g, i, m − a, b, c, g, i; G = d, e, f, m Step 4: Find the complement of H, denoted as H. H = U − H; H = a, b, c, d, e, f, g, i, m − c, d, e, g; H = a, b, f, i, m Step 5: Find the intersection of G and H, denoted as G ∩ H. G ∩ H = d, e, f, m ∩ a, b, f, i, m; G ∩ H = f, m Step 6: Compare the results from Step 2 and Step 5. Since G ∪ H = f, m and G ∩ H = f, m, Therefore, G ∪ H = G ∩ H is proven. d) What percent of the elements of universal set U have (G ∪ H ∪ K)? Step 1: Find the union of sets G, H, and K. G ∪ H ∪ K = a, b, c, g, i ∪ c, d, e, g ∪ f, g, i; G ∪ H ∪ K = a, b, c, d, e, f, g, i Step 2: Find the cardinal number of G ∪ H ∪ K. |G ∪ H ∪ 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 ∪ H ∪ K). Percentage = (|G ∪ H ∪ K|)/(|U|) × 100%; Percentage = 8/9 × 100% Percentage ≈ 0.8889 × 100% Percentage ≈ 88.89% The percentage of elements of universal set U that have (G ∪ H ∪ K) is approximately 88.89%.

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

Answers

Step-by-step solution

Part a· 2 steps

List all distinct elements in the universal set U from the Venn diagram.

Count the number of elements in U.

Part b· 3 steps
$\{g, i\}$

Identify the elements in set G.

Identify the elements in set K.

Find the intersection of G and K, which are the elements common to both sets.

Part c· 6 steps
: $\overline{G \cup H} = \overline{G} \cap \overline{H}$

Find $G \cup H$ .

Find the complement of $G \cup H$ , denoted as $\overline{G \cup H}$ .

Find the complement of G, denoted as $\overline{G}$ .

Find the complement of H, denoted as $\overline{H}$ .

Find the intersection of $\overline{G}$ and $\overline{H}$ , denoted as $\overline{G} \cap \overline{H}$ .

Compare the results from Step 2 and Step 5.

Part d· 4 steps
$88.89\%$

Find the union of sets G, H, and K.

Find the cardinal number of $G \cup H \cup K$ .

Recall the cardinal number of the universal set U.

Calculate the percentage of elements in $U$ that are in $(G \cup H \cup K)$ .

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More Mathematics Questions

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

Answers

Step-by-step solution

Part a· 2 steps

List all distinct elements in the universal set U from the Venn diagram.

Count the number of elements in U.

Part b· 3 steps{g,i}\{g, i\}{g,i}

Identify the elements in set G.

Identify the elements in set K.

Find the intersection of G and K, which are the elements common to both sets.

Part c· 6 steps: G∪H‾=G‾∩H‾\overline{G \cup H} = \overline{G} \cap \overline{H}G∪H=G∩H

Find G∪HG \cup HG∪H.

Find the complement of G∪HG \cup HG∪H, denoted as G∪H‾\overline{G \cup H}G∪H.

Find the complement of G, denoted as G‾\overline{G}G.

Find the complement of H, denoted as H‾\overline{H}H.

Find the intersection of G‾\overline{G}G and H‾\overline{H}H, denoted as G‾∩H‾\overline{G} \cap \overline{H}G∩H.

Compare the results from Step 2 and Step 5.

Part d· 4 steps88.89%88.89\%88.89%

Find the union of sets G, H, and K.

Find the cardinal number of G∪H∪KG \cup H \cup KG∪H∪K.

Recall the cardinal number of the universal set U.

Calculate the percentage of elements in UUU that are in (G∪H∪K)(G \cup H \cup K)(G∪H∪K).

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Part b· 3 steps
$\{g, i\}$

Part c· 6 steps
: $\overline{G \cup H} = \overline{G} \cap \overline{H}$

Find $G \cup H$ .

Find the complement of $G \cup H$ , denoted as $\overline{G \cup H}$ .

Find the complement of G, denoted as $\overline{G}$ .

Find the complement of H, denoted as $\overline{H}$ .

Find the intersection of $\overline{G}$ and $\overline{H}$ , denoted as $\overline{G} \cap \overline{H}$ .

Part d· 4 steps
$88.89\%$

Find the cardinal number of $G \cup H \cup K$ .

Calculate the percentage of elements in $U$ that are in $(G \cup H \cup K)$ .