Question

Determine which, if any, of the three statements are equivalent. Give a reason for your conclusion. Show complete work..
I) If the garden needs watering, then the garden does not need weeding.
II) Either the garden needs watering or the garden does not need weeding.
III) If the garden does not need watering, then the garden does not need weeding.

a. None are equivalent
b. I, II, and III are equivalent
c. I and II are equivalent
d. I and III are equivalent
e. II and III are equivalent

Answer 1

another one of these STUPID problems..lol

Answer 2

Basic steps are:

1) Translate each into symbolic form
2) Construct truth tables for each symbolic expression
3) Compare the last column of the truth table. If two tables have the same last column, then those corresponding expressions are equivalent

Answer 3

so lets call 1. p
2. q
3 r

Answer 4

right?

Answer 5

1. as in statement 1?

Answer 6

yes....

Answer 7

no need to be that broad, you can think smaller and say

p: the garden needs watering
q: the garden needs weeding.

Answer 8

oh ok.

Answer 9

so statement I for instance translates into

p --> ~q

Answer 10

ok so in words that would mean garden needs watering not if garden needs weeding? right

Answer 11

yes p --> ~q translates back into "If the garden needs watering, then the garden does not need weeding."

Answer 12

ok

Answer 13

how does the second one translate then...

Answer 14

p -->~q its the same thing

Answer 15

Key words "either" and "or" tell us to use a V and not a -->

So it translates to p V ~q

Answer 16

so 1 and 2 are not equivalent. right?

Answer 17

I didn't say that, but we can use either truth tables or logical simplification to find out

Answer 18

no

Answer 19

keep in mind that p --> q is the same as ~p V q

Answer 20

we have to figure out III.

Answer 21

so they are the same then?

Answer 22

i'm confused .. :( sorry

Answer 23

statement III translates to ~p --> ~q

Answer 24

ok i was starting to type that out for III.

Answer 25

and ~p --> ~q becomes ~(~p) V ~q which simplifies to p V ~q

Answer 26

oh sry

Answer 27

ok how come its not written out ~(~p) V ~ q? and written p V~q

Answer 28

because ~(~p) becomes just p

Answer 29

because its opposite right?

Answer 30

i know i'm a pain in the arse... sorry

Answer 31

it's ok, yes ~(~p) = p since ~ means opposite

Answer 32

~p means the opposite of p (~p = NOT p)

Answer 33

~(~p) is the opposite of ~p

~(~p) in english is NOT NOT p or just p

Answer 34

ok

Answer 35

im still with ya jim... :)

Answer 36

ok so what did you come up with

Answer 37

p V ~q

Answer 38

for statement III ok good, so this is the same as statement II

Answer 39

if you scroll back up or look at your previous work, you'll see this

Answer 40

so statements II and III are equivalent

Answer 41

yes...
so the answer would be B all three of them

Answer 42

no, because statement I is not the same as II or III

Answer 43

no it would be II and III same

Answer 44

yeah just II and III

Answer 45

ok... ty

Answer 46

yw