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

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