Question

Math

Posted 5 months ago

```
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) $\emptyset \subseteq \mathcal{P}(X)$
```

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Answer from Sia

Posted 5 months ago

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.

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