Let S be a set. Which one of the following is always correct? 1) a ∈ S and b ∈ S implies {a, b} ⊆ P(S) 2) a ∈ S and b ∈ S implies {a, b} ∈ P(S) 3) a ∈ S and b ∈ S implies {a, b} ∈... Let S be a set. Which one of the following is always correct? 1) a ∈ S and b ∈ S implies {a, b} ⊆ P(S) 2) a ∈ S and b ∈ S implies {a, b} ∈ P(S) 3) a ∈ S and b ∈ S implies {a, b} ∈ S 4) a ∈ S implies a ∈ P(S)
Understand the Problem
The question is asking to determine which of the given statements about set membership and power sets is always correct when certain conditions about elements a and b being in set S are met.
Answer
If $a \in S$ and $b \in S$, then $\{a\} \in \mathcal{P}(S)$ and $\{b\} \in \mathcal{P}(S)$.
Answer for screen readers
The statement that is always correct given the conditions is that if $a \in S$ and $b \in S$, then both ${a}$ and ${b}$ are elements of the power set $\mathcal{P}(S)$.
Steps to Solve
- Identify the Given Statements
List all the statements regarding set membership and power sets that need to be evaluated for correctness.
- Understanding Set Membership
Recall the definition of set membership. If $a \in S$, then $a$ is an element of set $S$. If $b \in S$, then the same applies for $b$.
- Understanding Power Sets
Recall that the power set of a set $S$, denoted as $\mathcal{P}(S)$, is the set of all subsets of $S$. This includes the empty set $\emptyset$, the set $S$ itself, and every combination of elements in $S$.
- Evaluating the Statements
For each statement, check if it logically follows from the properties of set membership and power sets. Formulate logical conclusions based on the definitions.
- Determine Which Statement is Always Correct
After evaluating each statement, identify the one that is universally true regardless of the specific elements involved, as long as the conditions about $a$ and $b$ being in $S$ are met.
The statement that is always correct given the conditions is that if $a \in S$ and $b \in S$, then both ${a}$ and ${b}$ are elements of the power set $\mathcal{P}(S)$.
More Information
This conclusion is based on the properties of subsets. Any single element of a set is always a subset of that set, hence it will always belong to its power set. A power set contains all possible subsets of the set.
Tips
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