Subset

### Is set \\(A = \{1, 2, 3\}\\) a subset of set \\(B = \{3, 2, 1, 0\}\\)? - [x] Yes, set \\(A\\) is a subset of set \\(B\\). - [ ] No, set \\(A\\) is not a subset of set \\(B\\). - [ ] It depends on the arrangement of elements in set \\(B\\). - [ ] Only if subsequent elements are added to \\(A\\). > **Explanation:** Set \\(A\\) is a subset of set \\(B\\) because all elements of \\(A\\) (i.e., 1, 2, and 3) are also in \\(B\\). The arrangement of elements doesn’t change this relationship. ### Which of the following represents the subset notation for \\(C = \{a, b\}\\) and \\(D = \{a, b, c\}\\)? - [x] \\(C \subseteq D\\) - [ ] \\(D \subseteq C\\) - [ ] \\(C \cap D = \emptyset\\) - [ ] \\(C \supseteq D\\) > **Explanation:** \\(C \subseteq D\\) correctly signifies that all elements of \\(C\\) (which are \\(a\\) and \\(b\\)) are contained within \\(D\\), making \\(C\\) a subset of \\(D\\). ### What does it mean if sets \\(E\\) and \\(F\\) are such that \\(E \subseteq F\\)? - [ ] Every element in \\(F\\) is also in \\(E\\). - [x] Every element in \\(E\\) is also in \\(F\\). - [ ] \\(E\\) and \\(F\\) have no common elements. - [ ] All elements in \\(E\\) are in \\(F\\) and vice versa. > **Explanation:** If \\(E \subseteq F\\), it means that every element in \\(E\\) is also in \\(F\\), which is the definition of a subset. ### Let \\(G = \emptyset\\). Can \\(\emptyset\\) be a proper subset of any set \\(H\\)? - [x] Yes, the empty set is a proper subset of any non-empty set \\(H\\). - [ ] No, \\(\emptyset\\) cannot be a subset. - [ ] Only if \\(H\\) also has finite elements. - [ ] Only if \\(H\\) is another empty set. > **Explanation:** The empty set \\(\emptyset\\) is considered a proper subset of any non-empty set because it contains no elements, fulfilling the condition trivially. ### Is it true that \\(\{1, 2\} \subseteq \{1, 2, 3, 4\}\\)? - [x] Yes, it is true. - [ ] No, it is false. - [ ] Only if the elements in \\(\{1, 2, 3, 4\}\\) are ordered differently. - [ ] Only for finite sets. > **Explanation:** \\(\{1, 2\}\\) is indeed a subset of \\(\{1, 2, 3, 4\}\\) because all elements of the first set are contained in the second set. ### Which of the following statements is incorrect? - [x] \\(\{3, 4\} \subset \{1, 2, 3\}\\) - [ ] \\(\{2, 3\} \subseteq \{2, 3\}\\) - [ ] \\(\emptyset \subseteq \{5, 6\}\\) - [ ] Every set is a subset of itself. > **Explanation:** \\(\{3, 4\}\\) cannot be a subset of \\(\{1, 2, 3\}\\) because the element 4 is not present in the latter set. ### Assume \\(X = \{a, b, c\}\\) and \\(Y = \{a, b, c, d\}\\). What best describes the relationship between \\(X\\) and \\(Y\\)? - [ ] \\(X\\) and \\(Y\\) are disjoint sets. - [x] \\(X \subseteq Y\\) - [ ] \\(Y \subseteq X\\) - [ ] \\(X \cap Y = \emptyset\\) > **Explanation:** Every element of \\(X\\) (\\(a, b, c\\)) is also an element of \\(Y\\), hence \\(X\\) is a subset of \\(Y\\). ### What does \\(P(A)\\) denote if \\(A\\) is a set? - [x] The power set of \\(A\\) - [ ] Proper subset of \\(A\\) - [ ] Parent set of \\(A\\) - [ ] Permutation of \\(A\\) > **Explanation:** \\(P(A)\\) denotes the power set of \\(A\\), which is the set containing all subsets of \\(A\\), including \\(A\\) itself and the empty set. ### Which of the following is always true for any set \\(Z\\)? - [x] \\(\emptyset \subseteq Z\\) - [ ] \\(Z \subset \emptyset\\) - [ ] \\(Z = \emptyset\\) - [ ] \\(Z \supseteq Z\\) > **Explanation:** The empty set is always a subset of any set by definition. ### If \\(R = \{3, 5, 7\}\\) and \\(S = \{7, 3, 5\}\\), are the sets \\(R\\) and \\(S\\) subsets of each other? - [x] True. - [ ] False. - [ ] Only if they have the same number of elements. - [ ] Only if listed in the same order. > **Explanation:** The sets \\(R\\) and \\(S\\) contain the same elements, merely ordered differently, hence each is a subset of the other, making them equal sets.