HW pgs 91-92 1-4 and then some

[Brandon Batchelor]

1)
a. What is the negation of [P or Q]? ~P&~Q
b. What is the negation of [P and ~Q]? ~PorQ
c. Using logic rules 3,4,and 5, show that P->Q means the same as [~P
or Q].

P->Q = ~[~[P->Q]] Rule 3
= ~[P^~Q] Rule 4
= [~PvQ] Rule 5

~PvQ = ~[~[~PvQ]] Rule 3
= ~[P^~Q] Rule 5
= ~[~[P->Q]] Rule 4
= P->Q Rule 3

2)Negate Euclid's Fourth Postulate.

Euclid's Fourth Postulate:
All right angles are congruent to each other.

Negation:
There exist some right angles not congruent to each other.

3)Negate the Euclidean Parallel Postulate.

Euclidean Parallel Postulate:
For every line L and for every point P that does not lie on L there
exists a unique line m through P that is parallel to L.

Negation:
For some line L and some point P that does not line on L, all lines
through P are not parallel to L.

4) State the converse of each of the following statements:

(a) If lines l and m are parallel, then a transversal t to lines l and
m cuts out congruent alternate interior angles.

Converse:
If a transversal t to lines l and m cuts out congruent alternate
interior angles, then lines l and m are parallel.

(b) If the sum of the degree measures of the interior angles on one
side of transversal t is less than 180 degrees, then lines l and m
meet on that side of transversal t.

Converse:
If lines l and m meet on one side of a transversal t, then the sum of
the degree measures of the interior angles that side of transversal t
is less than 180 degrees

Extra from my edition of the textbook:
Let S be the following self-referential statement: "Statement S is
false." Show that if S is either true or false then there is a
contradiction in our language. (This is the liar paradox. Kurt Godel
used a variant of it as the starting point for his famous
incompleteness theorem in logic; see DeLong, 1970)

A symbolic way of writing Rule 2 for RAA proofs is [[Hand ~C]->[S and
~S]->[H->C]. Explain this.