Question

1. if the conditional statement if i use my cell phone too much, my bill will be expensive. is true, which other statement must also be true? if my bill is expensive, i used my cell phone too much. if my bill is not expensive, i did not use my cell phone too much. if i do not use my cell phone too much, my bill will not be expensive. none must be true.

Explanation:

Let \(p\) be "I use my cell phone too much" and \(q\) be "my bill will be expensive". The given statement is \(p
ightarrow q\).

• The first option is \(q

ightarrow p\) (converse). A conditional statement \(p
ightarrow q\) does not imply its converse \(q
ightarrow p\). For example, there could be other reasons (like international calls) for the bill to be expensive.

• The second option is \(

eg q
ightarrow
eg p\) (contrapositive). By the law of contrapositive in logic, \(p
ightarrow q\equiv
eg q
ightarrow
eg p\). If the original statement \(p
ightarrow q\) (if \(p\) then \(q\)) is true, then its contrapositive (if not \(q\) then not \(p\)) is also true.

• The third option is \(

eg p
ightarrow
eg q\) (inverse). A conditional statement \(p
ightarrow q\) does not imply its inverse \(
eg p
ightarrow
eg q\). There could be cases where you use the phone moderately but due to extra - services the bill is expensive.

Answer:

If my bill is not expensive, I did not use my cell phone too much.