## princesspixie Group Title please tell me if this is correct: “All mathematicians must be good logicians and all good logicians must justify their claims.” one year ago one year ago

1. princesspixie Group Title

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. princesspixie Group Title

The Boxes are arrows

3. satellite73 Group Title

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. princesspixie Group Title

what would the r statement be?

5. satellite73 Group Title

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

6. princesspixie Group Title

would i say r: person justifies their claims

7. satellite73 Group Title

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

8. princesspixie Group Title

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

9. princesspixie Group Title

^ is the symbol for and?

10. satellite73 Group Title

yes

11. princesspixie Group Title

ok is the negation correct?

12. satellite73 Group Title

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. princesspixie Group Title

thats the negation ?

14. satellite73 Group Title

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

15. princesspixie Group Title

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

16. princesspixie Group Title

or infront of all the letters?

17. satellite73 Group Title

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

18. princesspixie Group Title

¬(p⟹q)∧(q⟹r) ?

19. princesspixie Group Title

what is the explanation for that ?

20. satellite73 Group Title

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

21. princesspixie Group Title

I dont understand..

22. satellite73 Group Title

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

23. princesspixie Group Title

No..

24. satellite73 Group Title

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

25. princesspixie Group Title

ok how do I explain the reasoning for it?

26. satellite73 Group Title

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

27. satellite73 Group Title

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. princesspixie Group Title

so both should be included in the answer?

29. satellite73 Group Title

yes but we have more work to do

30. princesspixie Group Title

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

31. satellite73 Group Title

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

32. satellite73 Group Title

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

33. princesspixie Group Title

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. satellite73 Group Title

actually i would get rid of the word "must"

35. satellite73 Group Title

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

36. princesspixie Group Title

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

37. satellite73 Group Title

$$\lnot$$ is the same as ~

38. satellite73 Group Title

means "not"

39. princesspixie Group Title

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

40. princesspixie Group Title

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

41. princesspixie Group Title

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. satellite73 Group Title

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. princesspixie Group Title

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

44. satellite73 Group Title

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. princesspixie Group Title

ok I understand

46. princesspixie Group Title

so would that be a good explanation?

47. satellite73 Group Title

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. princesspixie Group Title

49. satellite73 Group Title

i forget what C is

50. princesspixie Group Title

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

51. satellite73 Group Title

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. satellite73 Group Title

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

53. satellite73 Group Title

really to do this symbolically takes a while

54. princesspixie Group Title

ok I see

55. satellite73 Group Title

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

56. satellite73 Group Title

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. princesspixie Group Title

oh i see ok thanks!!

58. satellite73 Group Title

yw good luck with this

59. princesspixie Group Title

thanks so much for all your help!!