## anonymous 3 years ago Consider the following statement: “All mathematicians must be good logicians and all good logicians must justify their claims.” 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. b) Write the negation of this statement

1. anonymous

Okay, so suppose we have the form: "All p must be q". The question is does this mean \(p\implies q\) or \(q \implies p\)? You have to test it out a bit.

2. anonymous

So I'd start out with something like... "All dogs must be animals". Now does this mean "If you're a dog, then you're an animal." or does it mean "If you're an animal, you must be a dog"?

3. anonymous

@princesspixie Does this help?

4. anonymous

a little but what does a dis junction mean?

5. anonymous

Disjuntion isn't mentioned, but I'd assume it's two propositions combined with 'or'.

6. anonymous

oh i meant to say conjunction sorry!

7. anonymous

conjunction is when you take two propositions and combine them with and or or.

8. anonymous

but the sentence already has the word and in it so thats why im confused..

9. anonymous

Which just means you use and \( \wedge\) for it.

10. anonymous

??

11. anonymous

can you give me an example..

12. anonymous

this would be the first answer ? : “All mathematicians must be good logicians ^ good logicians must justify their claims.”

13. anonymous

No, because you need to convert them into conditionals.

14. anonymous

“if you are a mathematicians then you must be a good logician and all good logicians must justify their claims.” ??

15. anonymous

Yeah, now I'd say that \(p\) means 'person is mathematician' \(q\) means 'person is good logician' and \(r\) means 'justifies their claims'

16. anonymous

So we end up with \[ (p \implies q)\wedge(q \implies r) \]

17. anonymous

ok, is that part of the answer?

18. anonymous

That is part a.

19. anonymous

ok, great thanks! how would i negate it now? just throw a not in front of the original sentence?

20. anonymous

Yeah, throw in a \( \lnot \) .

21. ganeshie8

and you may have to simplify

22. ganeshie8

\( \neg ( ( p \implies q ) \wedge ( q \implies r ) ) \) \( \neg ( p \implies q ) \vee \neg ( q \implies r ) \) \( (p \wedge \neg q) \vee (q \wedge \neg r) \)

23. ganeshie8

which means, the statement is true, whenever a 'mathematician is not a logician' or whenever a 'logician is not a justfier'

24. ganeshie8

which forms a logical negation of the original statement

25. anonymous

ok thank you :)

26. ganeshie8

np.. yw :)

27. anonymous

so should the negation be : a mathematician is not a good logicians and logicians do not justify claims ?

28. anonymous

needs to be an or in the middle

29. anonymous

so would it be : a mathematician is not a good logician or logicians would justify their claims?

30. anonymous

@wio