SOLUTION: thank you. 6. Determine which, if any, of the three statements are equivalent. I) If the pipe is leaking, then I will not call the roofer. II) Either the pipe is leaking or I

Algebra.Com
Question 478096: thank you.
6. Determine which, if any, of the three statements are equivalent.
I) If the pipe is leaking, then I will not call the roofer.
II) Either the pipe is leaking or I will call the roofer.
III) If the pipe is not leaking, then I will call the roofer.

I and II are equivalent
II and III are equivalent
I and III are equivalent
I, II, and III are equivalent
None are equivalent

Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
this is a scary one but i think i have the answer.
statements are equivalent if they have the same truth value for all possible conditions.
you need to create a truth table for each statement.
the statements are:
I) If the pipe is leaking, then I will not call the roofer.
II) Either the pipe is leaking or I will call the roofer.
III) If the pipe is not leaking, then I will call the roofer.
the symbols we will use are:
p = the pipe is leaking
q = i will call the roofer.
converting the statements into symbols gets:
statement I becomes:
p->~q
statement II becomes:
(p)v(q)
statement III becomes:
~p->q
to see if the statements are equivalent, we use truth tables.
the truth tables for these statements are:
       p   q   ~p   ~q   p->~q   (p)v(q)   ~p->q
       T   T    F    F       F        T        T
       T   F    F    T       T        T        T
       F   T    T    F       T        T        T
       F   F    T    T       T        F        F

contrary to what you might believe otherwise, the truth table points to the fact that (p)v(q) is equivalent to ~p->q because their truth tables are the same under all conditions.
this makes the statements equivalent.
The answer to the question is that statements II and III are logically equivalent.
a reference on equivalent statements is shown below:
http://it.edgecombe.edu/homepage/killorant/MAT140/Module2/EquivalentStatements.pdf
-> means "if then"
v means "or"

RELATED QUESTIONS

Determine which, if any, of the three statements are equivalent. I) If the pipe is... (answered by lushis020910)
Determine which, if any, of the three statements are equivalent. I) If the weather is... (answered by richwmiller)
. Determine which, if any, of the three statements are equivalent. I) If there is not... (answered by Theo)
Which, if any, of the three statements are equivalent. I) If it is sunny, then I will... (answered by richard1234)
Determine which, if any, of the three statements are equivalent. I) If the rain is... (answered by richwmiller)
Determine which, if any, of the three statements are equivalent. I) If the dog wags its... (answered by richwmiller)
Determine which, if any, of the three statements are equivalent. I) If it is sunny... (answered by richwmiller)
Please help. Thank you! Determine which, if any, of the three statements are... (answered by richwmiller)
Determine which, if any, of the three statements are equivalent. a) If today is Monday, (answered by stanbon)