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. \( E=(s, m, a, r, t) \)

Certainly! Let's tackle each part of Exercise 2 step by step. For each set, we'll determine the number of possible subsets and then list (expand) those subsets. --- ### **1. Set \( A \): The word 'dOg'** **a. Determine the Set \( A \):** The word 'dOg' consists of the letters: - **d** - **O** - **g** Since sets contain **unique elements**, repeated letters are listed only once. Here, all letters are unique. \[ A = \{\, d,\ O,\ g \,\} \] **b. Number of Subsets:** For a set with \( n \) elements, the number of possible subsets is \( 2^n \). \[ n = 3 \Rightarrow 2^3 = 8 \text{ subsets} \] **c. Listing All Subsets of \( A \):** 1. **Empty Set:** \( \emptyset \) 2. **Single-element subsets:** - \( \{\, d \,\} \) - \( \{\, O \,\} \) - \( \{\, g \,\} \) 3. **Two-element subsets:** - \( \{\, d,\ O \,\} \) - \( \{\, d,\ g \,\} \) - \( \{\, O,\ g \,\} \) 4. **Three-element subset:** - \( \{\, d,\ O,\ g \,\} \) **_Total Subsets:_** 8 --- ### **2. Set \( B \): The word 'mirtor'** **a. Determine the Set \( B \):** The word 'mirtor' consists of the letters: - **m** - **i** - **r** - **t** - **o** - **r** Notice that the letter **r** appears twice. Since sets contain unique elements: \[ B = \{\, m,\ i,\ r,\ t,\ o \,\} \] **b. Number of Subsets:** \[ n = 5 \Rightarrow 2^5 = 32 \text{ subsets} \] **c. Listing All Subsets of \( B \):** Listing all 32 subsets here would be extensive, but they include: - The empty set \( \emptyset \) - All single-element subsets like \( \{\, m \,\} \), \( \{\, i \,\} \), etc. - All two-element subsets like \( \{\, m,\ i \,\} \), \( \{\, m,\ r \,\} \), etc. - Continuing up to the full set \( \{\, m,\ i,\ r,\ t,\ o \,\} \) **_Total Subsets:_** 32 --- ### **3. Set \( C \): The word 'paper'** **a. Determine the Set \( C \):** The word 'paper' consists of the letters: - **p** - **a** - **p** - **e** - **r** The letter **p** appears twice. So, the unique elements are: \[ C = \{\, p,\ a,\ e,\ r \,\} \] **b. Number of Subsets:** \[ n = 4 \Rightarrow 2^4 = 16 \text{ subsets} \] **c. Listing All Subsets of \( C \):** 1. **Empty Set:** \( \emptyset \) 2. **Single-element subsets:** - \( \{\, p \,\} \) - \( \{\, a \,\} \) - \( \{\, e \,\} \) - \( \{\, r \,\} \) 3. **Two-element subsets:** - \( \{\, p,\ a \,\} \) - \( \{\, p,\ e \,\} \) - \( \{\, p,\ r \,\} \) - \( \{\, a,\ e \,\} \) - \( \{\, a,\ r \,\} \) - \( \{\, e,\ r \,\} \) 4. **Three-element subsets:** - \( \{\, p,\ a,\ e \,\} \) - \( \{\, p,\ a,\ r \,\} \) - \( \{\, p,\ e,\ r \,\} \) - \( \{\, a,\ e,\ r \,\} \) 5. **Four-element subset:** - \( \{\, p,\ a,\ e,\ r \,\} \) **_Total Subsets:_** 16 --- ### **4. Set \( D = \{1, 2, 3, 4, 5, 6\} \)** **a. Determine the Set \( D \):** \[ D = \{\,1,\ 2,\ 3,\ 4,\ 5,\ 6\,\} \] **b. Number of Subsets:** \[ n = 6 \Rightarrow 2^6 = 64 \text{ subsets} \] **c. Listing All Subsets of \( D \):** Listing all 64 subsets is lengthy. Here's the structure: - **Empty Set:** \( \emptyset \) - **Single-element subsets:** 6 subsets (e.g., \( \{\,1\,\} \), \( \{\,2\,\} \), ...) - **Two-element subsets:** \( \binom{6}{2} = 15 \) subsets - **Three-element subsets:** \( \binom{6}{3} = 20 \) subsets - **Four-element subsets:** \( \binom{6}{4} = 15 \) subsets - **Five-element subsets:** \( \binom{6}{5} = 6 \) subsets - **Six-element subset:** \( \{\,1,\ 2,\ 3,\ 4,\ 5,\ 6\,\} \) **_Total Subsets:_** 64 --- ### **5. Set \( E = \{s, m, a, r, t\} \)** **a. Determine the Set \( E \):** \[ E = \{\, s,\ m,\ a,\ r,\ t \,\} \] **b. Number of Subsets:** \[ n = 5 \Rightarrow 2^5 = 32 \text{ subsets} \] **c. Listing All Subsets of \( E \):** Similar to Set \( B \), listing all 32 subsets here would be extensive. They include: - The empty set \( \emptyset \) - All single-element subsets like \( \{\, s \,\} \), \( \{\, m \,\} \), etc. - All two-element subsets like \( \{\, s,\ m \,\} \), \( \{\, s,\ a \,\} \), etc. - Continuing up to the full set \( \{\, s,\ m,\ a,\ r,\ t \,\} \) **_Total Subsets:_** 32 --- **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 practical to understand the pattern of subset formation rather than listing every single subset.