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# 2 logic questions.

1).

using only substitution and logical equivalences to verify

[ p -> (q ^ not q)] -> not p

is a tautology

*not has a symbol looks like 乛, but i just didnt know how to type that.

i could use the table method, but wonder how to verify it is a tautology with substitution and logical equivalences

2).provide steps and the reasons (for the steps) which establish the validity of the following argument.

p -> (q -> r)

p V s

t -> q

not s

______

not r -> not t

thx

### 2 Answers

- 1 decade agoFavorite Answer
Q1,

~ mean not

[ p -> (q ^ not q)] -> not p // *By Negation Laws,

[ p -> C ] -> ~p // *By Definition of -> , ( p-> C = ~p v C )

[~p v C] -> ~p // *By Identity Law , ( ~p v C = p )

[~p ] -> ~p // *By Definition of -> , (~p -> p = ~(~p) v ~p)

~(~p) v ~p // *By Double negation Law

p v ~p // *By Negation Laws

=> Tautology

Q2.

~s

p V s

p // by elimination

p -> ( q -> r )

q -> r // by modus ponens

t -> q

t -> r // by division in to cases

= ~r -> ~t // by contrapositive

Just curious, are you studying aboard ? I am in Washingtion State.

2008-09-25 00:38:16 補充：

p -> q and ~p v q have the same truth table,

therefore they are equivalent.

so ~p -> p have the same truth table of

~(~p) v p

2008-09-25 13:24:12 補充：

p -> q and ~p v q have the same truth table,

therefore they are equivalent.

so ~p -> p have the same truth table of

~(~p) v p

2008-09-28 02:31:16 補充：

the step can be move, you can move any step to prove it, the step doesn't matter.

2008-09-28 02:33:04 補充：

you can move arround the step, it is call rearraged, it would not change the conclusion,

step doesn't matter.

- 1 decade ago
thx, so i am just looking at the answer of Q.1

[~p ] -> ~p // *By Definition of -> , (~p -> p = ~(~p) v ~p) dont understand how to get

[~p ] -> to ~(~p) v

2008-09-24 12:26:19 補充：

so i have one step in the middle doesnt not understand.

and yes, i am studying aboard, i am in Victoria B.C.

thx and i will check on #2 later

2008-09-27 06:07:31 補充：

ok thx so i got Q.1

however Q2.

were u trying to moving around the order of the steps?

since the steps are fixed,

only steps, and reason can be added in between to make it explain the result.

thx

2008-09-28 06:00:51 補充：

got it, thx alot