Sure, let's analyze the problem step by step.
We are given the implication [tex]\( p \rightarrow q \)[/tex], where [tex]\( p \)[/tex] is "None of the oranges have been picked," and [tex]\( q \)[/tex] is "The oranges are ripe."
To find the contrapositive of this implication, we need to follow these steps:
1. State the Original Implication Using Given Statements:
[tex]\[
p \rightarrow q
\][/tex]
This translates to:
[tex]\[
\text{"If none of the oranges have been picked, then the oranges are ripe."}
\][/tex]
2. Form the Contrapositive of the Implication:
The contrapositive of an implication [tex]\( p \rightarrow q \)[/tex] is [tex]\( \neg q \rightarrow \neg p \)[/tex]. This means, "If [tex]\( q \)[/tex] is false, then [tex]\( p \)[/tex] is false."
3. Negate Each Statement:
- Negate [tex]\( q \)[/tex]: The negation of "The oranges are ripe" is "The oranges are not ripe."
- Negate [tex]\( p \)[/tex]: The negation of "None of the oranges have been picked" is "Some of the oranges have been picked."
4. Write the Contrapositive Using the Negated Statements:
Negating each part, we get:
[tex]\[
\neg q \rightarrow \neg p
\][/tex]
Which translates to:
[tex]\[
\text{"If the oranges are not ripe, then some of the oranges have been picked."}
\][/tex]
5. Choose the Correct Statement from Given Options:
We need to match our contrapositive statement with the given choices:
- Choice 1: "If the oranges are not ripe, then some of the oranges have been picked."
- Choice 2: "If some of the oranges have been picked, then the oranges are not ripe."
- Choice 3: "If none of the oranges have been picked, then the oranges are ripe."
- Choice 4: "If the oranges are ripe, then none of the oranges have been picked."
The correct match is Choice 1: "If the oranges are not ripe, then some of the oranges have been picked."
Thus, the contrapositive of [tex]\( p \rightarrow q \)[/tex] is correctly expressed by:
- Choice 1: "If the oranges are not ripe, then some of the oranges have been picked."
So the answer is:
[tex]\[ 1 \][/tex]
