Question

Discrete Math question...
will type equation, but I need to prove this using logical equivalences

Answer 1

\[[(p V q) \rightarrow r]\rightarrow[(p \rightarrow r) \Lambda (q \rightarrow r)]\]

Answer 2

Do you know what a truth table is?

Answer 3

vaguely

Answer 4

You have 3 variables: p,q, and r

To calculate the number of rows you will need for this truth table you do 2^(number of variables)

Then do all possible combinations for p,q, and r
and I mean like TTT, TTF, and so on...

Answer 5

Once you set that part we will look at each little expression inside the statement

Answer 6

so 8 rows

Answer 7

I mean like the P or Q thing
then the (P or Q) then r thing

then we will go to the other side of the implication which I think you actually meant to put an equivalent sign instead of ->

Answer 8

this is how the problem was typed out for me

Answer 9

right don't you agree there will be 8 possible combinations?
pqr
----
TTT
TTF
TFF
TFT
FTT
FTF
FFF
FFT

Answer 10

ahhh!!! yes!

Answer 11

good with that

Answer 12

now for each of those combinations decide if "P or Q" is T or F

Answer 13

We are doing this because this in the statement

Answer 14

sorry, that is where you are losing me...

Answer 15

and i don't mean to change the capital to lower case

you have in the table this already

pqr
----
TTT
TTF
TFF
TFT
FTT
FTF
FFF
FFT
===============================
now we are going to another column using what we already have in the rows to decide what p or q is
--------



pqr p or q
---- ______
TTT
TTF
TFF
TFT
FTT
FTF
FFF
FFT

Answer 16

for the first row you are given p and q are both true
so is p or q true?

Answer 17

can it be both?

Answer 18

http://en.wikipedia.org/wiki/Truth_table

check out this table
it gives you the definition (or logic) of the conjunctions and implications


this or is the inclusive or which means it can be both


the exclusive or will show an F when both are true

Answer 19

pqr p or q
---- ______
TTT T
TTF T
TFF T
TFT T
FTT T
FTF T
FFF T
FFT T

Answer 20

OOPS!

Answer 21

pqr p or q
---- ______
TTT T
TTF T
TFF T
TFT T
FTT T
FTF T
FFF F
FFT F

Answer 22

yeah great correction

Answer 23

now lets look at the whole left side of the implication

Answer 24

got going and then for got where to switch

Answer 25

ok

Answer 26

pqr p or q (p or q) then r
---- ______ -----------
TTT T
TTF T
TFF T
TFT T
FTT T
FTF T
FFF F
FFT F

Answer 27

you should see why we did one whole column for the p or q part now
we needed that column for this new column

Answer 28

pqr p or q (p or q) then r
---- ______ -----------
TTT T T
TTF T T
TFF T F
TFT T T
FTT T T
FTF T F
FFF F T?
FFT F F

Answer 29

do you know why the second row is wrong?

Answer 30

yes...

Answer 31

geez! sorry!

Answer 32

\[
\begin{array}{|c|c|c|c}
P
& Q
& R
& (P\lor{}Q)\to R \\
\hline
T & T & T & T \\
T & T & F & F \\
T & F & T &T \\
T & F & F & F \\
\hline
F & T & T & T \\
F & T & F & F \\
F & F & T & T \\
F & F & F &T \\
\hline
\end{array}
\]
do you know how long it took me to do this?
can i have a cookie?

Answer 33

you have a true if part and a false then part

Answer 34

it's okay @teach3.14
these truth tables can get long
and sometimes everyone including i make a mistake
seeing all these F and T make me dizzy and crazy

Answer 35

it looks very nice @satellite73

Answer 36

yes hes a show off

Answer 37

:)

Answer 38

just send me the cookie i will be on my way

Answer 39

cookie given

ok so now we need to go on the other side of the implication

Answer 40

and idea where to start there?

Answer 41

any*

Answer 42

\[p \rightarrow r\]

Answer 43

sounds great
you are working on the logic that we should do whats in the grouping symbols
great

Answer 44

ok so we need to make a new column for that

Answer 45

then a new column for q then r
then a new column for the whole right side

Answer 46

wait a sec, was his table correct? aren't lines 6 & 7 switched around for the (p v q) ->r?

Answer 47

let me take a look

Answer 48

looks like you are right about row 6 column 4

Answer 49

but everything else is right

Answer 50

pqr p or q (p or q) then r p then r q then r
---- ______ ----------- ___________ ---------
TTT T T T T
TTF T F F F
TFF T F F T
TFT T T T F
FTT T T F T
FTF T F T F
FFF F T T T
FFT F F F F

Answer 51

including the row 7 part

just the row 6 column 4

Answer 52

in his table*

Answer 53

ok good, I got worried for a second, thought I was following along!

Answer 54

5th and last line for the p then r
4th and last

Answer 55

if=false and then=true
then the if then statement is true

Answer 56

i know that is weird sounding
but that is logic

Answer 57

4th and last for q then r*

Answer 58

so the p and the q are like the if and the then?

Answer 59

and did you change values for the (p or q) then r thing?

Answer 60

for that column check out the 2nd row and last row

Answer 61

ok, I am afraid I am off on where we are now...

Answer 62

is this correct?
pqr p or q (p or q) then r p then r q then r
---- ______ ----------- ___________ ---------
TTT T T T T
TTF T F F F
TFF T F F T
TFT T T T T
FTT T T T T
FTF T F T F
FFF F T T T
FFT F F T T

Answer 63

ok look at the last row for (p or q) then r
and that is all i can see

Answer 64

pqr p or q (p or q) then r p then r q then r
---- ______ ----------- ___________ ---------
TTT T T T T
TTF T F F F
TFF T F F T
TFT T T T T
FTT T T T T
FTF T F T F
FFF F T T T
FFT T F T T

Answer 65

ok no....you change the wrong value

Answer 66

ok good, cause I didn't think that was right!

Answer 67

I was talking about the last row for (p or q) then r

Answer 68

pqr p or q (p or q) then r p then r q then r
---- ______ ----------- ___________ ---------
TTT T T T T
TTF T F F F
TFF T F F T
TFT T T T T
FTT T T T T
FTF T F T F
FFF F T T T
FFT F T T T

Answer 69

beautiful :)

Answer 70

now the last column

Answer 71

last column should be a pretty easy one
i love and

Answer 72

pqr p or q (p or q) then r p then r q then r p and q
---- ______ ----------- ___________ --------- _________
TTT T T T T T
TTF T F F F F
TFF T F F T F
TFT T T T T F
FTT T T T T F
FTF T F T F F
FFF F T T T T
FFT F T T T F

Answer 73

?

Answer 74

we don't care about p and q
we care about (p then r) and (q then r)

Answer 75

ahhh

Answer 76

for example
p then r is T in the first row
q then r is T is in the first row
so the and of those parts is also T in the first row

Answer 77

lol if you just want to write that column that is fine
because i think you have ran out of room

Answer 78

pqr p or q (p or q) then r p then r q then r p then r and q then r
---- ______ ----------- ___________ --------- ____________________
TTT T T T T T
TTF T F F F T
TFF T F F T F
TFT T T T T T
FTT T T T T T
FTF T F T F F
FFF F T T T T
FFT F T T T T

Answer 79

err i can't really read it
i'm sorry
but in order for an and statement to be true both parts must be true
all other cases write F

Answer 80

so ONLY if the are BOTH true?

Answer 81

you should notice something about the (p or q) then r column and also the (p then r) and (q then r) column
what do you know about each value for each row


right!

Answer 82

you can answer my question after you are done fixing the table

Answer 83

actually when you are done
i will probably restate me question to help better guide

Answer 84

pqr p or q (p or q) then r p then r q then r p then r and q then r
---- ______ ----------- _________ --------- _________________
TTT T T T T T
TTF T F F F F
TFF T F F T F
TFT T T T T T
FTT T T T T T
FTF T F T F F
FFF F T T T T
FFT F T T T T

Answer 85

I think I was actually correct

Answer 86

:)

Answer 87

so you must right down this whole true table that you have built (this work is for your instructor)

what i would do is highlight the boxes that we want to compare which is
(p or q) then r | (p then r) and (q then r)
T T
F F
F F
T T
T T
F F
T T
T T

Answer 88

they match

Answer 89

if you actually had a highlighter and i was you i would highlight these two columns
these columns are the important columns as they help us determine if the statements are logically equivalent

Answer 90

since they match they are logically equivalent

Answer 91

if they had one single mismatching and they are not logically equivalent and you would piss off any Vulcan

Answer 92

awesome! thanks!

Answer 93

you did great
you put in a lot of work
you deserve a medal also for all your hard work
you should try working some truth tables more too to get more practice

also remember
if you have and if then statement....
if=false and then=true --- the statement is true
if=false and then=false --- the statement is true

for "and statements" you need to just remember that you must have both parts true for the statement to be true

Answer 94

thank you for taking the time to walk me though that! I will try to practice some more!

Answer 95

and of course if you have:
if=true and then=true then the if then statement is totally true
I didn't say that above because that one i think is obvious

Answer 96

:)

Answer 97

have a goodnight
and good luck