Question

Read the two statements shown below.

If the weather is cold, Zakir will not go swimming.
The weather is not cold, or Zakir will not go swimming.

Create truth tables for the logical form of the two statements (not to be submitted). Use the truth tables to determine whether the two statements are logically equivalent. Justify your answer.

Answer 1

let,

p : the weather is cold
q : Zakir will not go swimming

Answer 2

then,
first statement can be written as: p->q
second statement is : ~p v q

Answer 3

right ?

Answer 4

Yes. Can you draw the table please ?

Answer 5

sure... but id like you to try.. let me correct if u go wrong... ?
do you know how to construct truth table ?

Answer 6

thats what i was going to do, i just need you to set it up

Answer 7

p q p->q ~p v q

T T X X
T F X X
F T X X
F F X X

Answer 8

thats the template for any truth table.
can u try filling the columns for p->q and ~p->q

Answer 9

what does " -> " mean ?

Answer 10

thats symbol means "implies"
its a conditional : "if, then"

Answer 11

and v ?

Answer 12

basic table for \(p\to q\)
\[\begin{array}{c|c|c}
p
& q
& p\to q \\
\hline
t & t & t \\
t & f & f \\
f & t & t \\
f & t & t \\

\end{array}\]

Answer 13

should memorize this one
only false if p is true and q is false

Answer 14

@NyKole are you familiar with the symbols
\[p\to q\]
\[p \land q\]
\[p\lor q\]
\[\lnot p\] etc?

Answer 15

no

Answer 16

then it will not be possible to do this problem
you need to know the symbols and the truth tables for each

Answer 17

for the p _> would it be

T
F
T
F

And ~ p v q would be

T
F
F
F
???

Answer 18

p q p->q ~p v q

T T T T
T F F F
F T T F
F F F F

Answer 19

a truth table is a table, not one column
i wrote the truth table for \(p\to q\) meaning "if p then q" or "p implies q" above

Answer 20

close but you have a mistake in the last line of \(p\to q\)

Answer 21

p q p->q ~p v q

T T T T
T F F F
F T T F
F F T F

Answer 22

ok good, now the \(p\to q\) part is correct

Answer 23

the \(\lnot p \lor q\) part is wrong
we need another step, we need to look at \(\lnot p\)

Answer 24

So that whole column is wrong ?

Answer 25

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

Answer 26

yes that column is wrong
for the last column \(\lnot p\lor q\) put a T if one or the other or both of \(\lnot p\) or \(q\) is true

Answer 27

for that one look only at the two columns \(\lnot p\) and \(q\)
if you see a T in either of those columns, put a T
if both are F, put an F

i can write it for you if you like

Answer 28

would it be
T
F
T
T

Answer 29

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

Answer 30

yes you have it!

Answer 31

and since both those columns are identical
the statements are logically equivalent

Answer 32

Ok yay. lol now for the second truth table for the question can you help me set it up also pease ?

Answer 33

which is the second one? we have them both in one table

Answer 34

for the second statement? how do you know they are logically equivalent ?

Answer 35

we have a column for \(p\to q\) and also a column for \(\lnot p \lor q\) so we are done

Answer 36

look at the column \(p\to q\) in the truth table and the column \(\lnot p \lor q\) we see that they are identical right?

Answer 37

yes. ok so like how would i justify they are logically equivalent for the question now ? like idk how i would explain it

Answer 38

no reason to make two completely different tables
you need only to compare the columns and see if they are the same or not
if they are the same, then the statements are equivalent
in other words, it is always true that \(p\to q\equiv \lnot p\lor q\)

Answer 39

i would explain it by saying "the column for \(p\to q\) and \(\lnot p\lor q\) are the same
therefore the statements are equivalent

Answer 40

it has to be at least 4 to 5 sentences. lol

Answer 41

and don't forget that although this is "logic" it should also make sense
so if you think about the statement
"if it is raining, i will get wet" and the statement
"it is not raining or i will get wet" are exactly the same

Answer 42

you can make up the sentences for yourself
i like being brief