Question

How to figure the truth table for the given: (p^r) right arrow (qvr)

Answer 1

We need to separate it into its components...
Like, those involving single variables first, like, p, q, and r.
Then, those involving single operators, like p^q, and qvr
And finally, work your way until ultimately, we get the last (whole) statement....

I understand that that may not have made much sense, so let's just demonstrate...

Answer 2

I was able to separate in to columns, just don't know if I have the "IF" column correct. May I share my answers with you from top to bottom and see if that is correct?

Answer 3

By all means... ^_^

Answer 4

T
T
F
F
F
F
T
T

Answer 5

Wait, this is the top row?

Answer 6

Your truth-table looks something like this, right?

Answer 7

\[\large \left[\begin{matrix}p&q&r&&p\land q&&q\lor r&&&(p\land q) \rightarrow(q\lor r)\\?&?&?&&?&&?&&&?\\?&?&?&&?&&?&&&?\\?&?&?&&?&&?&&&?\\?&?&?&&?&&?&&&?\\?&?&?&&?&&?&&&?\\?&?&?&&?&&?&&&?\\?&?&?&&?&&?&&&?\\?&?&?&&?&&?&&&?\end{matrix}\right]\]

Answer 8

Something like that. The answers I sent are what would go in the right arrow column.

Answer 9

I see... LOL
Well, unfortunately, it won't make sense to me because I don't know what goes in the first three columns (because what goes in the first three columns depends entirely on you)

So, is it like this?
\[\large \left[\begin{matrix}p&q&r&&p\land q&&q\lor r&&&(p\land q) \rightarrow(q\lor r)\\\color{green}T&\color{green}T&\color{green}T&&?&&?&&&?\\\color{green}T&\color{green}T&\color{red}F&&?&&?&&&?\\\color{green}T&\color{red}F&\color{green}T&&?&&?&&&?\\\color{green}T&\color{red}F&\color{red}F&&?&&?&&&?\\\color{red}F&\color{Green}T&\color{green}T&&?&&?&&&?\\\color{red}F&\color{green}T&\color{red}F&&?&&?&&&?\\\color{red}F&\color{red}F&\color{green}T&&?&&?&&&?\\\color{red}F&\color{red}F&\color{red}F&&?&&?&&&?\end{matrix}\right]\]

<pants>
points for effort lol


so... is that how your first three columns look like?

Answer 10

Yes, they do, so I'm on the right track!!! Yay. Now for the rest of it...

Answer 11

Might as well, yeah?
What do you have for the fourth column? :)

Answer 12

T
F
T
F
F
T
F
T

Answer 13

Fourth column? This one?
\[\large \left[\begin{matrix}p&q&r&&\color{blue}{p\land q}&&q\lor r&&&(p\land q) \rightarrow(q\lor r)\\\color{green}T&\color{green}T&\color{green}T&&\color{blue}?&&?&&&?\\\color{green}T&\color{green}T&\color{red}F&&\color{blue}?&&?&&&?\\\color{green}T&\color{red}F&\color{green}T&&\color{blue}?&&?&&&?\\\color{green}T&\color{red}F&\color{red}F&&\color{blue}?&&?&&&?\\\color{red}F&\color{Green}T&\color{green}T&&\color{blue}?&&?&&&?\\\color{red}F&\color{green}T&\color{red}F&&\color{blue}?&&?&&&?\\\color{red}F&\color{red}F&\color{green}T&&\color{blue}?&&?&&&?\\\color{red}F&\color{red}F&\color{red}F&&\color{blue}?&&?&&&?\end{matrix}\right]\]

Answer 14

I just realized that you have the problem wrong. Column 4 should be p^r

Answer 15

Whoops.. apologies... good thing I haven't filled in that table yet ^_^
\[\large \left[\begin{matrix}p&q&r&&\color{blue}{p\land r}&&q\lor r&&&(p\land r) \rightarrow(q\lor r)\\\color{green}T&\color{green}T&\color{green}T&&\color{blue}?&&?&&&?\\\color{green}T&\color{green}T&\color{red}F&&\color{blue}?&&?&&&?\\\color{green}T&\color{red}F&\color{green}T&&\color{blue}?&&?&&&?\\\color{green}T&\color{red}F&\color{red}F&&\color{blue}?&&?&&&?\\\color{red}F&\color{Green}T&\color{green}T&&\color{blue}?&&?&&&?\\\color{red}F&\color{green}T&\color{red}F&&\color{blue}?&&?&&&?\\\color{red}F&\color{red}F&\color{green}T&&\color{blue}?&&?&&&?\\\color{red}F&\color{red}F&\color{red}F&&\color{blue}?&&?&&&?\end{matrix}\right]\]

Answer 16

Sorry about that, just woke up XD

Answer 17

That's okay. I've got a massive migraine and want to make sure I completely understand this. And being that I am a visual learner, it really helps for someone to walk through this step by step with me. So I'm in no rush. Go splash some water on your face if you need to!!

Answer 18

But the column is still incorrect, however... keep in mind that we have a logical conjunction symbol \(\Large \land\) with the fourth column. Logical conjunction: fancy word for 'AND'.

So, it'll only be true for the rows where both p AND r are true.

And if I'm not mistaken, those are the first and third rows only...

Answer 19

Did that make sense? lol

Answer 20

Yes that did. Now with that in mind, let me figure the last column and I'll post in a quick sec.

Answer 21

Wait, just to be clear, may I know your revised fourth column?

Answer 22

Yes, it's:
T
F
T
F
F
F
F
F

Answer 23

Oh okay :D
What about the fifth column?

Answer 24

Okay, I know I'll need help on this one. I know that the V is "or" but I'm just going to shoot you my initial answers and we'll go from there:
T
F
F
T
T
F
F
T

Answer 25

Let me just fill in the fourth (for my own personal satisfaction :3 )
\[\large \left[\begin{matrix}p&q&r&&p\land r&&q\lor r&&&(p\land r) \rightarrow(q\lor r)\\\color{green}T&\color{green}T&\color{green}T&&\color{green}T&&?&&&?\\\color{green}T&\color{green}T&\color{red}F&&\color{red}F&&?&&&?\\\color{green}T&\color{red}F&\color{green}T&&\color{green}T&&?&&&?\\\color{green}T&\color{red}F&\color{red}F&&\color{red}F&&?&&&?\\\color{red}F&\color{Green}T&\color{green}T&&\color{red}F&&?&&&?\\\color{red}F&\color{green}T&\color{red}F&&\color{red}F&&?&&&?\\\color{red}F&\color{red}F&\color{green}T&&\color{red}F&&?&&&?\\\color{red}F&\color{red}F&\color{red}F&&\color{red}F&&?&&&?\end{matrix}\right]\]

Answer 26

Whatever floats your boat!

Answer 27

Now, as for the fifth, it's like the opposite of conjunction, it's disjunction...a fancy word for 'OR'.
if 'AND' is only true if both components are true...
'OR' is only *false* if BOTH components are *false*

Means it's true as long as at least one the components (q and r) is true.

Care to revise the column? :)

Answer 28

Yes,
T
T
T
F
T
T
T
F

Answer 29

You catch on really quickly :D

Answer 30

\[\large \left[\begin{matrix}p&q&r&&p\land r&&q\lor r&&&(p\land r) \rightarrow(q\lor r)\\\color{green}T&\color{green}T&\color{green}T&&\color{green}T&&\color{green}T&&&?\\\color{green}T&\color{green}T&\color{red}F&&\color{red}F&&\color{green}T&&&?\\\color{green}T&\color{red}F&\color{green}T&&\color{green}T&&\color{green}T&&&?\\\color{green}T&\color{red}F&\color{red}F&&\color{red}F&&\color{red}F&&&?\\\color{red}F&\color{Green}T&\color{green}T&&\color{red}F&&\color{green}T&&&?\\\color{red}F&\color{green}T&\color{red}F&&\color{red}F&&\color{green}T&&&?\\\color{red}F&\color{red}F&\color{green}T&&\color{red}F&&\color{green}T&&&?\\\color{red}F&\color{red}F&\color{red}F&&\color{red}F&&\color{red}F&&&?\end{matrix}\right]\]

Answer 31

Thanks! It helps when I can write it down too. I'm taking notes as we go along.

Answer 32

Finally, the implication... the one which isn't symmetrical...
This one is only false when the LEFT component is true BUT the right component is false..

It's true otherwise...
So, your revised sixth column is...? :D

Answer 33

Working on it...

Answer 34

Okay, so all of them are true!!!

Answer 35

If that is correct, I'd like to know why it is I keep on second guessing myself. I had all TRUE's the very first time I did it!

Answer 36

Yes, that is correct :)
\[\large \left[\begin{matrix}p&q&r&&p\land r&&q\lor r&&&(p\land r) \rightarrow(q\lor r)\\\color{green}T&\color{green}T&\color{green}T&&\color{green}T&&\color{green}T&&&\color{blue}T\\\color{green}T&\color{green}T&\color{red}F&&\color{red}F&&\color{green}T&&&\color{blue}T\\\color{green}T&\color{red}F&\color{green}T&&\color{green}T&&\color{green}T&&&\color{blue}T\\\color{green}T&\color{red}F&\color{red}F&&\color{red}F&&\color{red}F&&&\color{blue}T\\\color{red}F&\color{Green}T&\color{green}T&&\color{red}F&&\color{green}T&&&\color{blue}T\\\color{red}F&\color{green}T&\color{red}F&&\color{red}F&&\color{green}T&&&\color{blue}T\\\color{red}F&\color{red}F&\color{green}T&&\color{red}F&&\color{green}T&&&\color{blue}T\\\color{red}F&\color{red}F&\color{red}F&&\color{red}F&&\color{red}F&&&\color{blue}T\end{matrix}\right]\]

It can only mean that
\[\Large (p\land r) \implies (q\lor r)\]is a logical truth/tautology

Great job :D
<fireworks>
:3

Answer 37

It just boils down to knowledge of the basic truth tables, like for 'and', 'or' 'implies' etc :D

Answer 38

Thank you so much! I appreciate the tips and guidance! Talking me through it key. This is my very first truth table and wanted to make sure I was headed in the was at least headed in the right direction. I'm sure this won't be the first time I'll be tapping in to this site. This was awesome. Thanks again!!!

Answer 39

No problem :)
Welcome to Openstudy, and while you're at it, see if you can help out some people :D

Answer 40

I'll do my best!