1. Problem: Determine subset relations and equality between sets and elements as given. 2. Analyze each statement: - $C \subset D$: C is a subset of D. - $C \not\subset D$: C is not a subset of D. - $C = D$: C equals D. - $D \subset C$: D is a subset of C. 3. Check membership and subset relations: - Is $N \in S$? (Check if element N is in set S.) - Is $L \subset P$? (Check if L is a subset of P.) 4. Compare sets for equality: - $\{4,8,5,7\} = \{8,5,4,6\}$? No, because 7 \neq 6. - $\{16,10,30\} = \{10,16,30\}$? Yes, sets are equal regardless of order. - $\{51,14\} = \{14,51\}$? Yes, sets are equal. 5. Is $H$ a subset of the set? (Depends on definition of H and the set.) 6. Given: - $A = \{p,s,i,7\}$ - $B = \{s,i,m,p,a,7\}$ Check: - $B \subset A$? No, B has elements not in A (m, a). - $A \subset B$? Yes, all elements of A are in B. - $\{a\} \subset A$? No, a not in A. - $\{a\} \subset B$? Yes, a in B. - $\{m\} \not\subset A$? Correct, m not in A. - $\{s,p\} \subset B$? Yes, both in B. 7. For geometric parts: - a) Length $DC$ is a subset (segment) of triangle $\Delta ABD$. - b) Segment $CE$ is part of $\Delta ADC$. - c) $\Delta ADC \subset \Delta ABC$? No, ADC is a subtriangle of ABC only if points align properly. - d) Segment $EA \subset AD$? $EA$ is part of $AD$ if E lies on $AC$. - e) $\Delta ABD \subset \Delta ADC$? No, they share some points but are distinct triangles. - f) $DC \subset BC$? Yes, D lies on BC, so segment DC is part of BC. Final answers: - $C \subset D$: depends on sets. - $C \not\subset D$: depends. - $C = D$: depends. - $D \subset C$: depends. - $N \in S$: unknown without sets. - $L \subset P$: unknown. - $\{4,8,5,7\} \neq \{8,5,4,6\}$ - $\{16,10,30\} = \{10,16,30\}$ - $\{51,14\} = \{14,51\}$ - $B \not\subset A$ - $A \subset B$ - $\{a\} \not\subset A$ - $\{a\} \subset B$ - $\{m\} \not\subset A$ - $\{s,p\} \subset B$ - Geometric subset relations as above.