Given the two statements, determine the contrapositive of [tex]$p \rightarrow q$[/tex].

[tex]p:[/tex] None of the oranges have been picked.
[tex]q:[/tex] The oranges are ripe.

A. If the oranges are not ripe, then some of the oranges have been picked.
B. If some of the oranges have been picked, then the oranges are not ripe.
C. If none of the oranges have been picked, then the oranges are ripe.
D. If the oranges are ripe, then none of the oranges have been picked.



Answer :

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]