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"
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