Answer :
To solve the problem, we need to express the proposition [tex]\( r \rightarrow s \)[/tex] in an English sentence and determine whether it is true or false. Let's evaluate each proposition first.
Step 1: Evaluate proposition [tex]\( r \)[/tex]:
- Proposition [tex]\( r \)[/tex] states: [tex]\( 3^4 + 3^3 + 3^2 \)[/tex] is greater than 116.
- The expression [tex]\( 3^4 + 3^3 + 3^2 \)[/tex] evaluates to 117.
- Since 117 is greater than 116, proposition [tex]\( r \)[/tex] is true.
Step 2: Evaluate proposition [tex]\( s \)[/tex]:
- Proposition [tex]\( s \)[/tex] states: [tex]\( 1 \cdot 10^2 + 5 \cdot 10 + 2 \)[/tex] equals 116.
- The expression [tex]\( 1 \cdot 10^2 + 5 \cdot 10 + 2 \)[/tex] evaluates to 152.
- Since 152 is not equal to 116, proposition [tex]\( s \)[/tex] is false.
Step 3: Determine the logical statement [tex]\( r \rightarrow s \)[/tex]:
- A conditional statement [tex]\( r \rightarrow s \)[/tex] is expressed as "if [tex]\( r \)[/tex], then [tex]\( s \)[/tex]."
- In logical terms, a conditional statement [tex]\( r \rightarrow s \)[/tex] is true unless a true hypothesis (proposition [tex]\( r \)[/tex]) leads to a false conclusion (proposition [tex]\( s \)[/tex]).
Given that:
- Proposition [tex]\( r \)[/tex] is true.
- Proposition [tex]\( s \)[/tex] is false.
The conditional statement [tex]\( r \rightarrow s \)[/tex] is false, because a true hypothesis leading to a false conclusion makes the whole implication false.
Expressing in English Sentence:
Given our evaluations, the correct expression of the proposition [tex]\( r \rightarrow s \)[/tex] in an English sentence is:
B. If [tex]\( 3^4+3^3+3^2 \)[/tex] is greater than 116, then [tex]\( 1 \cdot 10^2+5 \cdot 10+2 \)[/tex] equals 116.
And, as concluded, this statement is false.
Step 1: Evaluate proposition [tex]\( r \)[/tex]:
- Proposition [tex]\( r \)[/tex] states: [tex]\( 3^4 + 3^3 + 3^2 \)[/tex] is greater than 116.
- The expression [tex]\( 3^4 + 3^3 + 3^2 \)[/tex] evaluates to 117.
- Since 117 is greater than 116, proposition [tex]\( r \)[/tex] is true.
Step 2: Evaluate proposition [tex]\( s \)[/tex]:
- Proposition [tex]\( s \)[/tex] states: [tex]\( 1 \cdot 10^2 + 5 \cdot 10 + 2 \)[/tex] equals 116.
- The expression [tex]\( 1 \cdot 10^2 + 5 \cdot 10 + 2 \)[/tex] evaluates to 152.
- Since 152 is not equal to 116, proposition [tex]\( s \)[/tex] is false.
Step 3: Determine the logical statement [tex]\( r \rightarrow s \)[/tex]:
- A conditional statement [tex]\( r \rightarrow s \)[/tex] is expressed as "if [tex]\( r \)[/tex], then [tex]\( s \)[/tex]."
- In logical terms, a conditional statement [tex]\( r \rightarrow s \)[/tex] is true unless a true hypothesis (proposition [tex]\( r \)[/tex]) leads to a false conclusion (proposition [tex]\( s \)[/tex]).
Given that:
- Proposition [tex]\( r \)[/tex] is true.
- Proposition [tex]\( s \)[/tex] is false.
The conditional statement [tex]\( r \rightarrow s \)[/tex] is false, because a true hypothesis leading to a false conclusion makes the whole implication false.
Expressing in English Sentence:
Given our evaluations, the correct expression of the proposition [tex]\( r \rightarrow s \)[/tex] in an English sentence is:
B. If [tex]\( 3^4+3^3+3^2 \)[/tex] is greater than 116, then [tex]\( 1 \cdot 10^2+5 \cdot 10+2 \)[/tex] equals 116.
And, as concluded, this statement is false.
