1. **Problem statement:** Given the set $A := \{2, 9, 12, 25, 42, 100\}$ and relations $R_1$ and $R_2$ defined on $A$, analyze membership and equivalence classes as described. 2. **Understanding the relations:** - $R_1$ and $R_2$ are relations on $A$. - We are given specific pairs and asked to verify their membership in $R$ or $R_1$, $R_2$. - Also, equivalence classes $[x]_R$ and their sizes $|[x]_R|$ are mentioned. 3. **Step b1:** Check if $(42, 2) \in R$ and $(42, 42) \notin R$. - Since $(42, 2) \in R_1$ is given, assume $R = R_1$ here. - $(42, 42) \notin R$ means $R$ is not reflexive at 42. 4. **Step b2:** Check if $(2, 42) \in R$. - No direct information given; if $R$ is not symmetric, $(2, 42)$ may or may not be in $R$. 5. **Step b3:** Given $(2, 9) \in R$, $(42, 2) \in R$, $(25, 42) \notin R$, and $|[25]_R| = 2$. - This suggests the equivalence class of 25 under $R$ has exactly 2 elements. 6. **For $R_2$:** - $(2, 9) \in R_2$, $12 \notin [2]_{R_2}$, and $|[42]_{R_2}| = 3$. - This means the equivalence class of 42 under $R_2$ has 3 elements. 7. **Summary:** - Relations $R$ and $R_2$ define different equivalence classes. - $R$ is not reflexive at 42. - $R_2$ has an equivalence class of size 3 containing 42. Since the problem is about verifying membership and class sizes, no explicit formula applies but understanding equivalence relations properties (reflexivity, symmetry, transitivity) is key. **Final answers:** - $(42, 2) \in R$ is true. - $(42, 42) \notin R$ is true. - $(2, 42) \in R$ is not confirmed. - $|[25]_R| = 2$. - $(2, 9) \in R_2$. - $12 \notin [2]_{R_2}$. - $|[42]_{R_2}| = 3$.