## princesspixie 2 years ago please tell me if this is correct: “All mathematicians must be good logicians and all good logicians must justify their claims.”

1. princesspixie

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

The Boxes are arrows

3. satellite73

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

what would the r statement be?

5. satellite73

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

6. princesspixie

would i say r: person justifies their claims

7. satellite73

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

8. princesspixie

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

9. princesspixie

^ is the symbol for and?

10. satellite73

yes

11. princesspixie

ok is the negation correct?

12. satellite73

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

thats the negation ?

14. satellite73

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

15. princesspixie

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

16. princesspixie

or infront of all the letters?

17. satellite73

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

18. princesspixie

¬(p⟹q)∧(q⟹r) ?

19. princesspixie

what is the explanation for that ?

20. satellite73

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

21. princesspixie

I dont understand..

22. satellite73

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

23. princesspixie

No..

24. satellite73

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

25. princesspixie

ok how do I explain the reasoning for it?

26. satellite73

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

27. satellite73

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

so both should be included in the answer?

29. satellite73

yes but we have more work to do

30. princesspixie

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

31. satellite73

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

32. satellite73

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

33. princesspixie

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

actually i would get rid of the word "must"

35. satellite73

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

36. princesspixie

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

37. satellite73

$$\lnot$$ is the same as ~

38. satellite73

means "not"

39. princesspixie

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

40. princesspixie

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

41. princesspixie

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

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

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

44. satellite73

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

ok I understand

46. princesspixie

so would that be a good explanation?

47. satellite73

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

thats the answer for C?

49. satellite73

i forget what C is

50. princesspixie

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

51. satellite73

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

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

53. satellite73

really to do this symbolically takes a while

54. princesspixie

ok I see

55. satellite73

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

56. satellite73

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

oh i see ok thanks!!

58. satellite73

yw good luck with this

59. princesspixie

thanks so much for all your help!!