Determine whether the following are true or false. Explain why shortly.
Let $X=\{1,2,\{3,4\},\{1\},\{5\}\}, Y=\{1,2,3\}$.
(a) $\{3,4,5\} \subseteq X$
(b) $\{1\} \in \mathcal{P}(X)$.
(c) $\{\{5\}\} \subseteq \mathcal{P}(X)$
(d) $\{\emptyset\} \in \mathcal{P}(\emptyset)$
(e) $\{1,2\} \in Y$.
(f) $\emptyset \subseteq \mathcal{P}(X)$

Question

Determine whether the following are true or false. Explain why shortly.
Let $X=\{1,2,\{3,4\},\{1\},\{5\}\}, Y=\{1,2,3\}$.
(a) $\{3,4,5\} \subseteq X$
(b) $\{1\} \in \mathcal{P}(X)$.
(c) $\{\{5\}\} \subseteq \mathcal{P}(X)$
(d) $\{\emptyset\} \in \mathcal{P}(\emptyset)$
(e) $\{1,2\} \in Y$.
(f) $\emptyset \subseteq \mathcal{P}(X)$

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ To determine if $\{3,4,5\} \subseteq X$, we check if every element of $\{3,4,5\}$ is also an element of $X$. ‖ ‖ step 2 ⋮ The set $X$ is given as $\{1,2,\{3,4\},\{1\},\{5\}\}$. ‖ ‖ step 3 ⋮ The elements 3 and 4 are contained within a nested set $\{3,4\}$ in $X$, not as individual elements, and 5 is contained within the nested set $\{5\}$. ‖ ‖ step 4 ⋮ Since $\{3,4,5\}$ are not individual elements of $X$, the statement $\{3,4,5\} \subseteq X$ is false. ‖ ∻Answer∻ ⚹ False ⚹ ∻Key Concept∻ ⚹Subset Relation⚹ ∻Explanation∻ ⚹A set A is a subset of set B if every element of A is also an element of B. Nested elements are not considered individual elements of the set.⚹ ∻Solution by Steps∻ ‖ step 1 ⋮ To determine if $\{1\} \in \mathcal{P}(X)$, we check if $\{1\}$ is an element of the power set of $X$. ‖ ‖ step 2 ⋮ The power set $\mathcal{P}(X)$ contains all subsets of $X$, including the individual elements as singleton sets. ‖ ‖ step 3 ⋮ Since $1$ is an element of $X$, the singleton set $\{1\}$ is a subset of $X$ and thus an element of $\mathcal{P}(X)$. ‖ ‖ step 4 ⋮ Therefore, the statement $\{1\} \in \mathcal{P}(X)$ is true. ‖ ∻Answer∻ ⚹ True ⚹ ∻Key Concept∻ ⚹Power Set and Elements⚹ ∻Explanation∻ ⚹The power set of a set contains all possible subsets of the original set, including the singleton sets of its elements.⚹ ∻Solution by Steps∻ ‖ step 1 ⋮ To determine if $\{\{5\}\} \subseteq \mathcal{P}(X)$, we check if the set containing the singleton set $\{5\}$ is a subset of the power set of $X$. ‖ ‖ step 2 ⋮ The power set $\mathcal{P}(X)$ contains all subsets of $X$, including the singleton set $\{5\}$. ‖ ‖ step 3 ⋮ Since $\{5\}$ is a subset of $X$, the set $\{\{5\}\}$, which contains $\{5\}$ as its only element, is a subset of $\mathcal{P}(X)$. ‖ ‖ step 4 ⋮ Therefore, the statement $\{\{5\}\} \subseteq \mathcal{P}(X)$ is true. ‖ ∻Answer∻ ⚹ True ⚹ ∻Key Concept∻ ⚹Subsets of Power Sets⚹ ∻Explanation∻ ⚹A set containing a subset of the original set is itself a subset of the power set of the original set.⚹ ∻Solution by Steps∻ ‖ step 1 ⋮ To determine if $\{\emptyset\} \in \mathcal{P}(\emptyset)$, we check if the set containing the empty set is an element of the power set of the empty set. ‖ ‖ step 2 ⋮ The power set of the empty set $\mathcal{P}(\emptyset)$ contains only two subsets: the empty set itself $\emptyset$ and the set containing the empty set $\{\emptyset\}$. ‖ ‖ step 3 ⋮ Therefore, the set $\{\emptyset\}$ is an element of $\mathcal{P}(\emptyset)$. ‖ ‖ step 4 ⋮ The statement $\{\emptyset\} \in \mathcal{P}(\emptyset)$ is true. ‖ ∻Answer∻ ⚹ True ⚹ ∻Key Concept∻ ⚹Power Set of the Empty Set⚹ ∻Explanation∻ ⚹The power set of the empty set contains the empty set itself and the set containing the empty set as its elements.⚹ ∻Solution by Steps∻ ‖ step 1 ⋮ To determine if $\{1,2\} \in Y$, we check if the set $\{1,2\}$ is an element of $Y$. ‖ ‖ step 2 ⋮ The set $Y$ is given as $\{1,2,3\}$. ‖ ‖ step 3 ⋮ The set $\{1,2\}$ is not an element of $Y$, but rather a subset of $Y$. ‖ ‖ step 4 ⋮ Therefore, the statement $\{1,2\} \in Y$ is false. ‖ ∻Answer∻ ⚹ False ⚹ ∻Key Concept∻ ⚹Element vs. Subset⚹ ∻Explanation∻ ⚹Being an element of a set and being a subset of a set are different; a set can be a subset without being an element.⚹ ∻Solution by Steps∻ ‖ step 1 ⋮ To determine if $\emptyset \subseteq \mathcal{P}(X)$, we check if the empty set is a subset of the power set of $X$. ‖ ‖ step 2 ⋮ The empty set $\emptyset$ is a subset of every set, including the power set of any set. ‖ ‖ step 3 ⋮ Therefore, the statement $\emptyset \subseteq \mathcal{P}(X)$ is true. ‖ ∻Answer∻ ⚹ True ⚹ ∻Key Concept∻ ⚹Empty Set as a Subset⚹ ∻Explanation∻ ⚹The empty set is universally a subset of any set, including the power set of any set.⚹◊Is it true that $Y \subseteq X$ based on the given sets $X$ and $Y$?⍭ Generate me a similar question◊

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