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please tell me if this is correct: “All mathematicians must be good logicians and all good logicians must justify their claims.”

Mathematics
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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
The Boxes are arrows
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

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Other answers:

what would the r statement be?
\(q\implies r\) : if a person is a good logician, then they justify their claims
would i say r: person justifies their claims
\[(p\implies q)\land (q\implies r)\] take those two statements and stick the word "and" between them
what do you mean those two statements and stick an and?
^ is the symbol for and?
yes
ok is the negation correct?
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
thats the negation ?
no that is \[(p\implies q)\land (q\implies r)\]
ok would I see.. would I put a ~ before the q and the r to negate it ?
or infront of all the letters?
no you have to put \(\lnot\) in front of the whole thing
¬(p⟹q)∧(q⟹r) ?
what is the explanation for that ?
\[\lnot\left((p\implies q)\land (q\implies r)\right)\] then you should rewrite without the \(\implies \) and use demorgan laws
I dont understand..
it will take a while do you know that \(p\implies q\equiv \lnot p\lor q\)?
No..
to negate a statement you put a big \(\lnot\) in front of the whole thing, you do not negate each piece separately
ok how do I explain the reasoning for it?
you have some statement "blah blah blah" the the negation is "it is not true that blah blah blah"
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 \[\lnot \left((\lnot p \lor q)\land (\lnot q\lor r)\right)\]
so both should be included in the answer?
yes but we have more work to do
Im still not sure how I should explain the answer ..
unless you are unfamiliar with these laws. perhaps you are supposed to do something else
your original statement All mathematicians must be good logicians and all good logicians must justify their claims is correct
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
actually i would get rid of the word "must"
"all mathematicians are good logicians and all good logicians justify their claims" perhaps is better
thats why i thought this symbol would be in the equation ~
\(\lnot\) is the same as ~
means "not"
ok so should i use the ~ if that what the text book uses?
ok .. how should i explain that in simple terms?
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.
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
ok so i should use that instead of if a person is not a good... ?
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"
ok I understand
so would that be a good explanation?
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
thats the answer for C?
i forget what C is
c) Now use part b) to write the negation of this statement verbally.
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
the negation of the "and" statement is one or the other is false
really to do this symbolically takes a while
ok I see
\[\lnot \left((\lnot p \lor q)\land (\lnot q\lor r)\right)\] \[\lnot(\lnot p\lor q)\lor \lnot(\lnot q \lor r)\] \[(p\land \lnot q)\lor (q\land \lnot r)\]
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
oh i see ok thanks!!
yw good luck with this
thanks so much for all your help!!

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