## satellite73 4 years ago show $[(p\rightarrow q)\land (q\rightarrow r)]\rightarrow (p\rightarrow r)$ is a tautology, without using truth tables

1. anonymous

Seems like transitivity.

2. amistre64

-[(p>q)n(q>r)] v (p>r) -(p>q) v -(q>r) v (p>r) -(-pvq) v -(-qvr) v (-pvr) (pn-q) v (qn-r) v -p v r

3. anonymous

it is entirely obvious, but somehow i get messed up at the last step

4. anonymous

ok so i am fine at $(p\land \lnot q)\lor (q\land \lnot r)\lor (\lnot p \land r)$

5. amistre64

(pn-q) v (qn-r) v-p vr -------------- (pv-p) n (-qv-p) v (qvr) n (-rvr) T n (-qv-p) v (qvr) n T (Tn-q) v (Tn-p) v (qnT) v (rnT) -q v-p v q v r (-qvq) v-p v r T v ..... = T

6. anonymous

Can't you just write that it represents a syllogism?

7. anonymous

@amistre, you lost me on the last line

8. anonymous

yeah i am trying to do it amistre way

9. amistre64

since we got all ors, and a T; the rest dont matter

10. amistre64

T or ? or ? or ? = T

11. anonymous

that is fine , but what is this (pn-q) v (qn-r) v-p vr ?

12. amistre64

oring -p to the first bit and r to the last bit i stacked them vertical; helps me see whats going on

13. anonymous

ok that is the line i need to get starting here $(p\land \lnot q)\lor (q\land \lnot r)\lor (\lnot p \land r)$

14. amistre64

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

15. amistre64

a -> changes to "v" not "n"

16. amistre64

p -> r :: -p v r

17. anonymous

ah ok so we have $(p\land \lnot q)\lor (q\land \lnot r)\lor (\lnot p \lor r)$ and then $(p\land \lnot q)\lor (q\land \lnot r)\lor \lnot p \lor r$ and then we are going to commute to $\lnot p \lor (p \land \lnot q) \lor (q \land \lnot r) \lor q$

18. amistre64

yep

19. amistre64

not a vq, vr

20. amistre64

good, and from there its cake walk; hot coals cake walk, but cake walk nonetheless :)

21. anonymous

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

22. anonymous

i think that is right now yes?

23. amistre64

yep; then we can T off the sides and distribute thru the middles which cancels out the ts

24. anonymous

god this makes my head spin. thanks

25. amistre64

yeah, i blame guass lol

26. anonymous

well i tried to do this on the fly and got hopelessly stuck, then i tried it again and got stuck again not on the fly. thanks for straightening it out!