## anonymous 4 years ago please tell me if this is correct: “All mathematicians must be good logicians and all good logicians must justify their claims.”

1. 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

2. anonymous

The Boxes are arrows

3. 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

4. anonymous

what would the r statement be?

5. anonymous

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

6. anonymous

would i say r: person justifies their claims

7. anonymous

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

8. anonymous

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

9. anonymous

^ is the symbol for and?

10. anonymous

yes

11. anonymous

ok is the negation correct?

12. 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

13. anonymous

thats the negation ?

14. anonymous

no that is $(p\implies q)\land (q\implies r)$

15. anonymous

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

16. anonymous

or infront of all the letters?

17. anonymous

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

18. anonymous

¬(p⟹q)∧(q⟹r) ?

19. anonymous

what is the explanation for that ?

20. anonymous

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

21. anonymous

I dont understand..

22. anonymous

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

23. anonymous

No..

24. anonymous

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

25. anonymous

ok how do I explain the reasoning for it?

26. anonymous

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

27. 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 $\lnot \left((\lnot p \lor q)\land (\lnot q\lor r)\right)$

28. anonymous

so both should be included in the answer?

29. anonymous

yes but we have more work to do

30. anonymous

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

31. anonymous

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

32. anonymous

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

33. 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

34. anonymous

actually i would get rid of the word "must"

35. anonymous

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

36. anonymous

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

37. anonymous

$$\lnot$$ is the same as ~

38. anonymous

means "not"

39. anonymous

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

40. anonymous

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

41. 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.

42. 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

43. anonymous

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

44. 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"

45. anonymous

ok I understand

46. anonymous

so would that be a good explanation?

47. 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

48. anonymous

49. anonymous

i forget what C is

50. anonymous

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

51. 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

52. anonymous

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

53. anonymous

really to do this symbolically takes a while

54. anonymous

ok I see

55. anonymous

$\lnot \left((\lnot p \lor q)\land (\lnot q\lor r)\right)$ $\lnot(\lnot p\lor q)\lor \lnot(\lnot q \lor r)$ $(p\land \lnot q)\lor (q\land \lnot r)$

56. 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

57. anonymous

oh i see ok thanks!!

58. anonymous

yw good luck with this

59. anonymous

thanks so much for all your help!!