Exercise 2 Direction: Given the following items below, identify the number of subsets then expand. 1. Let \( A \) be the set of the word ' \( \mathrm{dOg}^{\prime} \) '. 2. Let \( B \) be the set of the word 'mirtor'. 3. Let \( C \) be the set of the word 'paper'. 4. \( D=\{1,2,3,4,5,6\} \) 5. \( E=(s, m, a, r, t) \)

Sure! Let's analyze each set provided, determine the number of subsets for each, and then list (expand) those subsets. ### General Principle: For any set with \( n \) elements, the number of possible subsets is \( 2^n \). This includes the empty set and the set itself. --- ### 1. Set \( A \) - The word "dOg" **a. Define the Set:** - The word "dOg" consists of the characters: 'd', 'O', 'g'. - Therefore, \( A = \{\, d,\, O,\, g \,\} \) **b. Number of Subsets:** - \( n = 3 \) - Number of subsets \( = 2^3 = 8 \) **c. Subsets of \( A \):** 1. \( \emptyset \) (Empty Set) 2. \( \{\, d \,\} \) 3. \( \{\, O \,\} \) 4. \( \{\, g \,\} \) 5. \( \{\, d,\, O \,\} \) 6. \( \{\, d,\, g \,\} \) 7. \( \{\, O,\, g \,\} \) 8. \( \{\, d,\, O,\, g \,\} \) --- ### 2. Set \( B \) - The word "mirtor" **a. Define the Set:** - The word "mirtor" consists of the letters: 'm', 'i', 'r', 't', 'o', 'r'. - Since sets contain unique elements, duplicate 'r' is considered once. - Therefore, \( B = \{\, m,\, i,\, r,\, t,\, o \,\} \) **b. Number of Subsets:** - \( n = 5 \) - Number of subsets \( = 2^5 = 32 \) **c. Subsets of \( B \):** Listing all 32 subsets can be extensive, but here's how they are structured: 1. \( \emptyset \) 2. Single-element subsets: \( \{\, m \,\} \), \( \{\, i \,\} \), \( \{\, r \,\} \), \( \{\, t \,\} \), \( \{\, o \,\} \) 3. Two-element subsets: All combinations like \( \{\, m, i \,\} \), \( \{\, m, r \,\} \), ..., \( \{\, t, o \,\} \) 4. Three-element subsets: All combinations like \( \{\, m, i, r \,\} \), ..., \( \{\, r, t, o \,\} \) 5. Four-element subsets: All combinations like \( \{\, m, i, r, t \,\} \), ..., \( \{\, i, r, t, o \,\} \) 6. The entire set: \( \{\, m, i, r, t, o \,\} \) *(For brevity, only the structure is shown. Each subset can be explicitly listed as per requirement.)* --- ### 3. Set \( C \) - The word "paper" **a. Define the Set:** - The word "paper" consists of the letters: 'p', 'a', 'p', 'e', 'r'. - Removing duplicates, we get: 'p', 'a', 'e', 'r'. - Therefore, \( C = \{\, p,\, a,\, e,\, r \,\} \) **b. Number of Subsets:** - \( n = 4 \) - Number of subsets \( = 2^4 = 16 \) **c. Subsets of \( C \):** 1. \( \emptyset \) 2. \( \{\, p \,\} \) 3. \( \{\, a \,\} \) 4. \( \{\, e \,\} \) 5. \( \{\, r \,\} \) 6. \( \{\, p, a \,\} \) 7. \( \{\, p, e \,\} \) 8. \( \{\, p, r \,\} \) 9. \( \{\, a, e \,\} \) 10. \( \{\, a, r \,\} \) 11. \( \{\, e, r \,\} \) 12. \( \{\, p, a, e \,\} \) 13. \( \{\, p, a, r \,\} \) 14. \( \{\, p, e, r \,\} \) 15. \( \{\, a, e, r \,\} \) 16. \( \{\, p, a, e, r \,\} \) --- ### 4. Set \( D \) - The set \( \{1, 2, 3, 4, 5, 6\} \) **a. Define the Set:** - \( D = \{\, 1,\, 2,\, 3,\, 4,\, 5,\, 6 \,\} \) **b. Number of Subsets:** - \( n = 6 \) - Number of subsets \( = 2^6 = 64 \) **c. Subsets of \( D \):** Listing all 64 subsets is extensive, but they include: 1. \( \emptyset \) 2. All single-element subsets: \( \{1\} \), \( \{2\} \), ..., \( \{6\} \) 3. All two-element subsets: \( \{1,2\} \), \( \{1,3\} \), ..., \( \{5,6\} \) 4. All three-element subsets: \( \{1,2,3\} \), ..., \( \{4,5,6\} \) 5. All four-element subsets: \( \{1,2,3,4\} \), ..., \( \{3,4,5,6\} \) 6. All five-element subsets: \( \{1,2,3,4,5\} \), ..., \( \{2,3,4,5,6\} \) 7. The entire set: \( \{1,2,3,4,5,6\} \) *(Each combination can be explicitly listed as needed.)* --- ### 5. Set \( E \) - The tuple \( (s, m, a, r, t) \) **a. Define the Set:** - Although presented as a tuple, to form a set, we consider unique elements. - \( E = \{\, s,\, m,\, a,\, r,\, t \,\} \) **b. Number of Subsets:** - \( n = 5 \) - Number of subsets \( = 2^5 = 32 \) **c. Subsets of \( E \):** 1. \( \emptyset \) 2. Single-element subsets: \( \{s\} \), \( \{m\} \), \( \{a\} \), \( \{r\} \), \( \{t\} \) 3. Two-element subsets: \( \{s, m\} \), \( \{s, a\} \), ..., \( \{r, t\} \) 4. Three-element subsets: \( \{s, m, a\} \), ..., \( \{a, r, t\} \) 5. Four-element subsets: \( \{s, m, a, r\} \), ..., \( \{m, a, r, t\} \) 6. The entire set: \( \{s, m, a, r, t\} \) *(Each subset can be explicitly listed as required.)* --- ### Summary Table: | Set | Elements | Number of Subsets | |-----|----------|-------------------| | **A** | \( \{\, d,\, O,\, g \,\} \) | 8 | | **B** | \( \{\, m,\, i,\, r,\, t,\, o \,\} \) | 32 | | **C** | \( \{\, p,\, a,\, e,\, r \,\} \) | 16 | | **D** | \( \{\, 1,\, 2,\, 3,\, 4,\, 5,\, 6 \,\} \) | 64 | | **E** | \( \{\, s,\, m,\, a,\, r,\, t \,\} \) | 32 | --- **Note:** For larger sets like \( D \) and \( E \), it's often impractical to list all subsets manually. However, understanding the structure and being able to compute the total number is essential.