1. **Problem statement:** Given the universal set $$\varepsilon = \{x : x \text{ is an odd integer and } 1 \leq x \leq 25\}$$, and sets $$X = \{x : x \text{ is a multiple of } 3\}$$ and $$Y = \{x : x \text{ is a multiple of } 5\}$$, find the members of: a. $$X \cap Y$$ b. $$X \cap Y'$$ 2. **Understanding the sets:** - $$\varepsilon$$ contains all odd integers from 1 to 25: $$\{1,3,5,7,9,11,13,15,17,19,21,23,25\}$$. - $$X$$ is the subset of $$\varepsilon$$ with elements that are multiples of 3. - $$Y$$ is the subset of $$\varepsilon$$ with elements that are multiples of 5. - $$Y'$$ is the complement of $$Y$$ in $$\varepsilon$$, i.e., elements in $$\varepsilon$$ not in $$Y$$. 3. **Find elements of $$X$$:** Multiples of 3 in $$\varepsilon$$ are odd multiples of 3 between 1 and 25: $$3, 9, 15, 21$$ So, $$X = \{3, 9, 15, 21\}$$. 4. **Find elements of $$Y$$:** Multiples of 5 in $$\varepsilon$$ are odd multiples of 5 between 1 and 25: $$5, 15, 25$$ So, $$Y = \{5, 15, 25\}$$. 5. **Find $$X \cap Y$$:** Intersection means elements common to both sets: $$X \cap Y = \{15\}$$ 6. **Find $$Y'$$:** Complement of $$Y$$ in $$\varepsilon$$ is all elements in $$\varepsilon$$ not in $$Y$$: $$Y' = \varepsilon - Y = \{1,3,7,9,11,13,15,17,19,21,23,25\} - \{5,15,25\} = \{1,3,7,9,11,13,17,19,21,23\}$$ 7. **Find $$X \cap Y'$$:** Elements in $$X$$ that are also in $$Y'$$: $$X = \{3,9,15,21\}$$ and $$Y' = \{1,3,7,9,11,13,17,19,21,23\}$$ Common elements are $$3, 9, 21$$ **Final answers:** a. $$X \cap Y = \{15\}$$ b. $$X \cap Y' = \{3, 9, 21\}$$