Let's identify the converse of the statement [tex]\( R \rightarrow S \)[/tex], which is "If it rains, then he will stay home."
To start, let's recall that in logic, the converse of an implication [tex]\( P \rightarrow Q \)[/tex] is the statement [tex]\( Q \rightarrow P \)[/tex].
Given the statement [tex]\( R \rightarrow S \)[/tex]:
- [tex]\( R \)[/tex]: "It rains"
- [tex]\( S \)[/tex]: "He will stay home"
The converse of [tex]\( R \rightarrow S \)[/tex] is formed by switching the hypothesis and the conclusion. So, the converse will be [tex]\( S \rightarrow R \)[/tex].
- [tex]\( S \rightarrow R \)[/tex]: "If he stays home, then it will rain."
Let's match this with the provided options:
A) [tex]\( S \rightarrow R \)[/tex]: "If he stays home, then it will rain."
B) [tex]\( \sim R \rightarrow \sim S \)[/tex]: This is the inverse of the original statement. "If it doesn't rain, then he won't stay home."
C) [tex]\( R \backsim S \)[/tex]: This represents a biconditional statement (if and only if), which means "It will rain if and only if he stays home."
D) [tex]\( \sim R \rightarrow S \)[/tex]: This is another implication with a different hypothesis. "If it doesn't rain, then he will stay home."
Therefore, the correct answer is:
A) [tex]\( S \rightarrow R \)[/tex]: "If he stays home, then it will rain."
So the final conclusion is:
[tex]\[ \boxed{1} \][/tex]
