truth, like gold, is to be obtained not by its growth, but by washing away from it all that is not...
Post on 26-Dec-2015
214 Views
Preview:
TRANSCRIPT
Truth, like gold, is to be obtained not by its growth, but by washing away from it all that is not gold. ~Leo Tolstoy
We discussed before that equivalent statements have the same truth values in all cases (see last two columns). There are different ways to achieve equivalent statements.
P Q ~P ~Q PQ ~Q~P
T T F F T T
T F F T F F
F T T F T T
F F T T T T
If a conditional is stated as “if p, then q” then here are the representations of the various forms of that conditional:
Conditional pq if p, then q Converse qp if q, then p Inverse ~p~q if not p, then not q Contrapositive ~q~p if not q, then not p
◦
This is the truth table for all four forms of a given conditional. Notice which columns have the same truth values (and are therefore equivalent).
P Q ~P ~Q PQ QP
~P~Q
~Q~P
T T F F T T T T
T F F T F T T F
F T T F T F F T
F F T T T T T T
In a truth table, a negation is obvious because it has the opposite truth values of the original statement in every case.
It is also true that the negation of a negation would yield the original statement (and its truth values)
~ (~ P)≡P
An if-then statement is only false when you have a true premise, and a false conclusion (p is true, q is false). Therefore, to negate a conditional you would use that statement (P^~Q).
P Q ~Q PQ P^~Q
T T F T F
T F T F T
F T F T F
F F T T F
DeMorgan’s Laws can apply here for creating equivalent statements.
~ (P∧Q) ≡~P∨~Q
~ (P∨Q) ≡~P∧~Q
: (p∧q) =~p∨~q~(p∨q) =~p∧~q
: (p∧q) =~p∨~q~(p∨q) =~p∧~q
In order to create equivalent statements, you may need to make use of the negation, converse, inverse, contrapositive, or DeMorgan’s Laws.
You can always use a truth table to verify if two statements are equivalent or negations of each other.
3.5, p. 136-138; #4-14 even, 24-34 even
top related