Question

Please Help! Will fan and medal! Read the proposition and the implication shown below.
"If the lights are off, there is no one inside."

Which statement is a contrapositive to the proposition and logically equivalent to it? side.

A. If the lights are not off, there is someone in.
B. If there is no one inside, the lights will not be on.
C. If the lights are on, there is someone inside.
D. If there is someone inside, the lights will be on.

Answer 1

If p then q.
Contrapositive: If not q, then not p.

Answer 2

I think it's c but I'm not sure

Answer 3

I'll show it to you step by step.
Conditional: \(If \color{red}{~the ~lights ~are ~off},~ [then]~\color{blue}{there ~is ~no ~one ~inside.}"\)

The above is your given conditional.

Answer 4

okay thanks

Answer 5

The red part is the hypothesis, and the blue part is the conclusion.
The "if" part is the hypothesis.
The "then" part is the conclusion.
Still ok?

Answer 6

okay

Answer 7

A typical generic conditional statement is:

Conditional: \(If ~\color{red}{hypothesis},~ then~\color{blue}{conclusion.}"\)

In symbols, it is:

Conditional: \(If ~\color{red}{p},~ then~\color{blue}{q.}"\)

where p stands for the hypothesis, and q stands for the conclusion.

Answer 8

Keep the following in mind:
p is the hypothesis
q is the conclusion
Conditional: If p, then q.

Answer 9

What is the contrapositive?
For a conditional, if p then q, the contrapositive is
If not q, then not p.

Answer 10

Conditional: \(If ~\color{red}{hypothesis},~ then~\color{blue}{conclusion.}"\)

Contrapositive: \(If ~opposite~ of~ the~ \color{blue}{conclusion.}, ~then ~opposite~of~the~\color{red}{hypothesis}"\)

Answer 11

Conditional: If p, then q.
Contrapositive: If not q, then not p.

Answer 12

In other words, if you start with a conditional, to find the contrapositive you need to do this:
1. switch the hypothesis and the conclusion
2. deny the hypothesis and the conclusion

Answer 13

Back to your problem.

Conditional: "If the lights are off, there is no one inside."

To find the contrapositive:
1. switch the hypothesis and the conclusion

If there is no inside, then the lights are off.

Now deny both the hypothesis and the conclusion.

If there is someone inside, then the lights are on.

The last statement above is the contrapositive.