Equivalence

Equivalent statement forms always have the same truth value, and we assert that fact by expressing the equivalence in the form of a biconditional. Such statements have the same truth value on every line of the corresponding truth table.

The importance of this point for our purposes is that equivalent statements can be substituted for each other in a proof.

Tautology (Taut)
Double Negation (DN)
p (p p)
p ~ ~ p
  (p v p)
Commutation (Com)
Association (Assoc)
(p q) (q p)
[p (q r)] [(p q) r]
(p v q) (q v p)
[p v (q v r)] [(p v q) v r]
Distribution (Dist)
De Morgan's Law (DM)
[p (q v r)] [(p q) v (p r)]
~ (p q) (~ p v ~ q)
[p v (q r)] [(p v q) (p v r)]
~ (p v q) (~ p ~ q)
Contraposition (Contra)
Implication (Imp)
(p q) (~ q ~ p)
(p q) (~ p v q)
     ~ (p ~ q)
Biconditional (Bicon)
Exportation (Exp)
(p q) [(p q) (q p)]
[(p q) r] [p (q r)]
    [(p q) v (~ p ~ q)]


Comprehension Questions

1 Identify the following equivalence: p (p v p).
a) tautology
b) commutation
c) distribution
d) contraposition
e) double negation
2 Identify the following equivalence: [(p . q) > r] [p > (q > r)].
a) association
b) De Morgan's law
c) implication
d) biconditional
e) exportation
3 Identify the following equivalence: ~(p . q) ( ~p v ~q).
a) association
b) De Morgan's law
c) implication
d) biconditional
e) exportation
4 Identify the following equivalence: [p v (q v r)] [(p v q) v r].
a) association
b) De Morgan's law
c) implication
d) biconditional
e) exportation
5 Identify the following equivalence: (p v q) (q v p).
a) tautology
b) commutation
c) distribution
d) contraposition
e) double negation
6 Identify the following equivalence: (p > q) (~q > ~p).
a) tautology
b) commutation
c) distribution
d) contraposition
e) double negation
7 Identify the following equivalence: p ~~p.
a) tautology
b) commutation
c) distribution
d) contraposition
e) double negation
8 Identify the following equivalence: (p > q) (~p v q).
a) association
b) De Morgan's law
c) implication
d) biconditional
e) exportation
9 Identify the following equivalence: [p . (q v r)] [(p . q) v (p . r)].
a) tautology
b) commutation
c) distribution
d) contraposition
e) double negation
10 Identify the following equivalence: (p . q) (q . p).
a) tautology
b) commutation
c) distribution
d) contraposition
e) double negation
11 Identify the following equivalence: ~(p v q) (~p . ~q).
a) association
b) De Morgan's law
c) implication
d) biconditional
e) exportation
12 Identify the following equivalence: (p = q) [(p > q).(q > p)].
a) association
b) De Morgan's law
c) implication
d) biconditional
e) exportation
13 Identify the following equivalence: [p.(q.r)] [(p.q).r].
a) association
b) De Morgan's law
c) implication
d) biconditional
e) exportation
14 Identify the following equivalence: [p v (q . r)] [(p v q).(p v r)] .
a) tautology
b) commutation
c) distribution
d) contraposition
e) double negation
15 Identify the following equivalence: (p = q) [(p . q) v (~p . ~q)].
a) association
b) De Morgan's law
c) implication
d) biconditional
e) exportation
16 Identify the following equivalence: (p > q) ~(p .~ q) .
a) association
b) De Morgan's law
c) implication
d) biconditional
e) exportation


Conditional Proof and Reductio ad Absurdum

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