Question

please tell me if this is correct:


“All mathematicians must be good logicians and all good logicians must justify their claims.”

Answer 1

a) Write this statement symbolically as a conjunction of two conditional statements. Use three components (p, q, and r) and explicitly state what these are in your work. Remember: a component in logic is a simple declarative sentence that is either true or false. For example, "You are a student in MAT 101" would be a component statement. Note how this sentence contains no logical connectives like "not", "or", "and", etc.
b) Write the negation of this statement symbolically. Explain your reasoning here.
c) Now use part b) to write the negation of this statement verbally.

A- p means 'person is mathematician' q means 'person is good logician' and r means 'justifies their claims'
(p⟹q)∧(q⟹r)


B- (~p⟹~q)∧(~q⟹~r)
You put the ~ mark because that negates the statement.

C - If a person is not a mathematician then the person is not a good logician or they do not justify their claims

Answer 2

The Boxes are arrows

Answer 3

p: a person is a good mathematician
q: a person is a good logician

\(p\implies q\) : if a person is a good mathematician, then a person is a good logician
rather tortured english, would probably say
if a person is a good mathematician then he or she is also a good logician

Answer 4

what would the r statement be?

Answer 5

\(q\implies r\) :
if a person is a good logician, then they justify their claims

Answer 6

would i say
r: person justifies their claims

Answer 7

\[(p\implies q)\land (q\implies r)\] take those two statements and stick the word "and" between them

Answer 8

what do you mean those two statements and stick an and?

Answer 9

^ is the symbol for and?

Answer 10

yes

Answer 11

ok is the negation correct?

Answer 12

if a person is a good logician, then they justify their claims and if a person is a good logician, then they justify their claims

Answer 13

thats the negation ?

Answer 14

no that is \[(p\implies q)\land (q\implies r)\]

Answer 15

ok would I see.. would I put a ~ before the q and the r to negate it ?

Answer 16

or infront of all the letters?

Answer 17

no you have to put \(\lnot\) in front of the whole thing

Answer 18

¬(p⟹q)∧(q⟹r) ?

Answer 19

what is the explanation for that ?

Answer 20

\[\lnot\left((p\implies q)\land (q\implies r)\right)\] then you should rewrite without the \(\implies \) and use demorgan laws

Answer 21

I dont understand..

Answer 22

it will take a while
do you know that \(p\implies q\equiv \lnot p\lor q\)?

Answer 23

No..

Answer 24

to negate a statement you put a big \(\lnot\) in front of the whole thing, you do not negate each piece separately

Answer 25

ok how do I explain the reasoning for it?

Answer 26

you have some statement "blah blah blah" the the negation is "it is not true that blah blah blah"

Answer 27

if you want to negate it you have to rewrite it as
\[(\lnot p \lor q)\land (\lnot q\lor r)\] then negate it as
\[\lnot \left((\lnot p \lor q)\land (\lnot q\lor r)\right)\]

Answer 28

so both should be included in the answer?

Answer 29

yes but we have more work to do

Answer 30

Im still not sure how I should explain the answer ..

Answer 31

unless you are unfamiliar with these laws. perhaps you are supposed to do something else

Answer 32

your original statement
All mathematicians must be good logicians and all good logicians must justify their claims
is correct

Answer 33

ive never seen that little hook thing, but i see that in the textbook it says converse inverse and contrapostive .. not sure if that helps

Answer 34

actually i would get rid of the word "must"

Answer 35

"all mathematicians are good logicians and all good logicians justify their claims" perhaps is better

Answer 36

thats why i thought this symbol would be in the equation ~

Answer 37

\(\lnot\) is the same as ~

Answer 38

means "not"

Answer 39

ok so should i use the ~ if that what the text book uses?

Answer 40

ok .. how should i explain that in simple terms?

Answer 41

is this good... The symbol ~ means not. Therefore, if a person is not a good mathematician then the person is not a good logician, and the person cannot justify their claims.

Answer 42

i would think perhaps the negation, without using the symbols, would be
not all mathematicians are good logicians or not all logicians justify their claims

Answer 43

ok so i should use that instead of if a person is not a good... ?

Answer 44

if a person is a good mathematician, then they are a good logician
the statement
if a person is not a good mathematician then the person is not a good logician
is not the negation, it is the "inverse"

Answer 45

ok I understand

Answer 46

so would that be a good explanation?

Answer 47

then negation of "if a person is a good mathematician, then they are a good logician" is
there is some mathematician who is not a good logician

Answer 48

thats the answer for C?

Answer 49

i forget what C is

Answer 50

c) Now use part b) to write the negation of this statement verbally.

Answer 51

the negation of the whole thing would read something like
"there is a good mathematician who is not a good logician, or there is a good logician who does not justify their claims

Answer 52

the negation of the "and" statement is one or the other is false

Answer 53

really to do this symbolically takes a while

Answer 54

ok I see

Answer 55

\[\lnot \left((\lnot p \lor q)\land (\lnot q\lor r)\right)\]
\[\lnot(\lnot p\lor q)\lor \lnot(\lnot q \lor r)\]
\[(p\land \lnot q)\lor (q\land \lnot r)\]

Answer 56

the last statement says there is a good mathematician who is not a good logician or there is a good logician who does not justify their claims

Answer 57

oh i see ok thanks!!

Answer 58

yw good luck with this

Answer 59

thanks so much for all your help!!