Question

construct and complete a truth table with the following headings:

1) p
2) q
3) p → q
4) p ∧ (p → q)
5) [p ∧ (p → q)] → q

Answer 1

we can do this

Answer 2

if you are here that is, it is not that hard

Answer 3

ready?

Answer 4

yes

Answer 5

I have no idea how to do this lol @satellite73

Answer 6

ok first we start here
\[\begin{array}{c|c}
p
& q \\
\hline
& \\
& \\
& \\
& \\

\end{array}\]

Answer 7

then we put all combinations of t and f underneath
usually like this \[\begin{array}{|c|c}
p
& q \\
\hline
T & T \\
T & F \\
F & T \\
F & F \\
\hline
\end{array}\]

Answer 8

that that each possible pair is there
have you seen something like this before?

Answer 9

next step is \[\begin{array}{|c|c|c}
p
& q
& p\to q \\
\hline
T & T & \\
T & F & \\
F & T & \\
F & F & \\
\hline
\end{array}\]

Answer 10

\(p\to q\) is always true, unless \(p\) is true and \(q\) is false
it will look like this \[\begin{array}{|c|c|c}
p
& q
& p\to q \\
\hline
T & T &T \\
T & F & F\\
F & T & T\\
F & F & T\\
\hline
\end{array}\]

Answer 11

how we doing so far? are you totally lost, or is it okay ?

Answer 12

oh okay im starting to get it

Answer 13

the first two columns are there to get all possible combinations of \(T\) and\(F\) for \(p,q\)

Answer 14

the third column \(p\to q\) as i said is always true unless \(p\) is \(T\) and \(q\) is \(F\)

Answer 15

as your read across
two more to go

Answer 16

This is what the project says:
On a sheet of paper, construct and complete a truth table with the following headings:

p q p → q p ∧ (p → q) [p ∧ (p → q)] → q
2. Construct and complete a truth table in order to determine if the following statements are logically equivalent:

p → q and ~q → ~p
3. Explain in the box below if the statement [p ∧ (p → q)] → q is a tautology or not.

4. Explain in the box below if the statements p → q and ~q → ~p are logically equivalent or not.

5. Turn your paper with your truth tables in to your teacher.

Answer 17

\[\begin{array}{|c|c|c|c|}
p
& q
& p\to q
&p\land (p\to q) \\
\hline
T & T &T & \\
T & F & F&\\
F & T & T&\\
F & F & T&\\
\hline
\end{array}\]

Answer 18

we will get there if you have the patience for it
we are not there yet

Answer 19

we have to fill in the column \(p\land (p\to q)\)

Answer 20

for this column you only look at the column under \(p\) and the column under \(p\to q\)
the "and" statement is only true if there is a T in both rows

Answer 21

\[\begin{array}{|c|c|c|c|}
p
& q
& p\to q
&p\land (p\to q) \\
\hline
T & T &T &T \\
T & F & F&F\\
F & T & T&F\\
F & F & T&F\\
\hline
\end{array}\]

Answer 22

if i have totally lost you let me know
or if you have a specific question please ask
we have one more column to fill

Answer 23

What does the ^ mean

Answer 24

\(p\land q\) means "\(p\) AND \(q\)"
it is only True if both \(p\) and \(q\) are True

Answer 25

like saying "it is raining and cold" is only true if it is both raining and cold

Answer 26

okay but i dont get what the last column means all together. p ∧ (p → q)

Answer 27

in english it is kind of funny
it says "p and p implies q"

Answer 28

oh okay

Answer 29

if we change p to "it is sunny" and q to "i go to the beach" it says
"it is sunny and if it is sunny i will go to the beach"

Answer 30

oh okay!

Answer 31

last column ready? it is going to take me a minute to format it

Answer 32

okay ready

Answer 33

\[\begin{array}{|c|c|c|c|c}
p
& q
& p\to q
&p\land (p\to q)
&[p\land (p\to q)]\to q \\
\hline
T & T &T &T& \\
T & F & F&F&\\
F & T & T&F&\\
F & F & T&F&\\
\hline
\end{array}\]

Answer 34

the reason they have you do one at a time is so we can understand the last one
the last one is the \([p\land (p\to q)]\to q\) which is always true unless \(p\land (p\to q)\) is true and \(q\) is false
so we look at the column under \(p\land (p\to q)\) and \(q\) and if we see \(T, F\) we put \(F\) otherwise we put \(T\)

Answer 35

but if you look carefully, you see that it is never the case
if \(p\land (p\to q)]\) is \(T\) then \(q\) is also \(T\) in that row \[\begin{array}{|c|c|c|c|c}
p
& q
& p\to q
&p\land (p\to q)
&[p\land (p\to q)]\to q \\
\hline
T &\color{red} T &T &\color{red}T& \\
T & F & F&F&\\
F & T & T&F&\\
F & F & T&F&\\
\hline
\end{array}\]

Answer 36

therefore the last column gets all T\[\begin{array}{|c|c|c|c|c}
p
& q
& p\to q
&p\land (p\to q)
&[p\land (p\to q)]\to q \\
\hline
T & T &T &T& T\\
T & F & F&F&T\\
F & T & T&F&T\\
F & F & T&F&T\\
\hline
\end{array}\]

Answer 37

now we can answer question 3 as well
since the last column has all \(T\) that means it is always true, and is a "tautology"

Answer 38

back to the previous example, the last column says
"it is sunny, and if it is sunny i go to the beach, then i go to the beach"
pretty obviously true right?

Answer 39

yeah so they are equivalent?

Answer 40

hold the phone
you need two different statements to ask "are they equivalent"
we were answering this question
3. Explain in the box below if the statement [p ∧ (p → q)] → q is a tautology or not.

Answer 41

oh i thought were on 4 lol

Answer 42

the answer to that one is "yes, it is a tautology"

Answer 43

#4 requires doing #2, which is a lot easier than the last one

Answer 44

okay how do i figure it out

Answer 45

ok here we go

Answer 46

\[\begin{array}{|c|c|c|c|c|c}
p
& q
& \lnot{}p
& \lnot{}q
& p\to q
& \lnot{}q\to\lnot{}p \\
\hline
T & T & & & & \\
T & F & & & & \\
F & T & & & & \\
F & F & & & & \\
\hline
\end{array}\]

Answer 47

as before, first two columns give all combinations of T and F
now to get \(\lnot p\) and \(\lnot q\) change all T to F and F to T

Answer 48

do you know what \(\lnot p\) means?

Answer 49

isnt it just the opposite of p?

Answer 50

\[\begin{array}{|c|c|c|c|c|c}
p
& q
& \lnot{}p
& \lnot{}q
& p\to q
& \lnot{}q\to\lnot{}p \\
\hline
T & T & F &F \\
T & F &F & T& & \\
F & T & T& F & & \\
F & F & T& T& \\
\hline
\end{array}\]
yes it means "not p"

Answer 51

you see how i got the column \(\lnot p\)? i looked at the column for \(p\) and if i see T i put F and if i see F i put T
same for \(\lnot q\)

Answer 52

now we already did \(p\to q\) for the last question
it is always T unless \(p\) is T and \(q\) is F, in the second row

Answer 53

\[\begin{array}{|c|c|c|c|c|c}
p
& q
& \lnot{}p
& \lnot{}q
& p\to q
& \lnot{}q\to\lnot{}p \\
\hline
T & T & F &F &T \\
T & F &F & T&F & \\
F & T & T& F & T& \\
F & F & T& T& T \\
\hline
\end{array}\]

Answer 54

and finally we look at \(\lnot q\to \lnot p\) which is always true unless \(\lnot q\) it T and \(\lnot p\) is F

Answer 55

that only occurs here \[\begin{array}{|c|c|c|c|c|c}
p
& q
& \lnot{}p
& \lnot{}q
& p\to q
& \lnot{}q\to\lnot{}p \\
\hline
T & T & F &F &T \\
T & F &\color{red}F & \color{red}T&F & \\
F & T & T& F & T& \\
F & F & T& T& T \\
\hline
\end{array}\]

Answer 56

fill it in like this \[\begin{array}{|c|c|c|c|c|c}
p
& q
& \lnot{}p
& \lnot{}q
& p\to q
& \lnot{}q\to\lnot{}p \\
\hline
T & T & F &F &T&T \\
T & F &\color{red}F & \color{red}T&F &F \\
F & T & T& F & T& T \\
F & F & T& T& T &T\\
\hline
\end{array}\]

Answer 57

now we look at the last two columns and see that they are identical
guess what that means?

Answer 58

they are equivalent!

Answer 59

exactly
back to our english example
if it is sunny, i will go to the beach
if i did not go to the beach, then it is not sunny
they say the same thing

Answer 60

Oh okay could you help me set up the chart?

Answer 61

@satellite73

Answer 62

set up what chart?

Answer 63

Oh nvm you already did i think for number 2 it says to set up the truth table

Answer 64

lordamercy
what do you think we were doing for the past hour?!

Answer 65

lol but which chart is the one that proves they are equivalent

Answer 66

the entire table is the truth table for both
the fact that the last two columns are identical is what tells you the two statements are equivalent

Answer 67

\[\begin{array}{|c|c|c|c|c|c}
p
& q
& \lnot{}p
& \lnot{}q
& p\to q
& \lnot{}q\to\lnot{}p \\
\hline
T & T & F &F &T&T \\
T & F &F & T&F &F \\
F & T & T& F & T& T \\
F & F & T& T& T &T\\
\hline
\end{array}\]this is the truth table for both \(p\to q\) and \(\lnot q\to \lnot p\)
under both, you see the same thing