Definitions
The CONJUNCTION of two propositions p and q is written p ∧ q. It denotes “p and q”.
The DISJUNCTION of two propositions p and q is written p ∨ q. It denotes “p or q”.
The CONDITIONAL PROPOSITION is written p → q, is the proposition “if p then q”.
p is the HYPOTHESIS or PREMISE and q is the CONCLUSION or CONSEQUENCE
p and q are EQUIVALENT and write p ≡ q if they always have the same truth value.
The BICONDITIONAL PROPOSITION p ↔ q is “p if, and only if, q”. It is true when p and q have the same truth values
and is false otherwise (exactly the same)
Initial
p→q
≡
￢p ∨ q
Contrapositive
￢q →￢p
≡
￢p ∨ q
Converse
q→p
≡
￢p ∨ q
Inverse
￢p → ￢q
≡
￢p ∨ q
Commutative laws:
p ∧ q ≡ q ∧ p;
p∨q≡q∨p
Associative laws:
(p ∧ q) ∧ r ≡ p ∧ (q ∧ r); (p ∨ q) ∨ r ≡ p ∨ (q ∨ r)
Distributive laws:
p ∧ (q ∨ r) ≡ (p ∧ q) ∨ (p ∧ r)
p ∨ (q ∧ r) ≡ (p ∨ q) ∧ (p ∨ r)
Identity laws:
p ∧ T ≡ p;
p∨C≡p
p ∧￢p = C;
p ∨￢p = T
Negation laws:
Double negation laws:
￢(￢p) ≡ p
Idempotent laws:
p ∧ p ≡ p;
p∨p≡p
p ∧ C ≡ C;
p∨T≡T
￢(p ∧ q) ≡ ￢p ∨ ￢q;
￢(p ∨ q) ≡ ￢p ∧ ￢q
Universal bound laws:
De Morgan’s laws:
Absorption laws:
p ∧ (p ∨ q) ≡ p; p ∨ (p ∧ q) ≡ p
Negations of T and C:
￢C ≡ T;
￢T ≡ C
We imagine the actions suggested by the quantifiers as being performed in the order in which the quantifiers occur. Reversing
the order of the quantifiers can change the truth value of a statement unless the quantifiers are all universal or all existential.
For example, “everyone loves someone” is translated into:
∀x∃y P(x,y) where P(x,y) is “x loves y.”
However
∃y∀x P(x,y)
is “there is someone who is loved by everyone”.
∀
∀
∃
∃
x∃
y∃
x∀
y∀
y P(x, y) – For all x, there exist y such that P is true
x P(x, y) – For all y, there exist x such that P is true
y P(x, y) – There exist x such that for all y, P is true
x P(x, y) – There exist y such that for all x, P is true
To Prove: p → q
Direct Proof:
Start by assuming that p is true. Subsequent steps are constructed using various rules of inference, with the final step showing
that q is true.
Proof by Contraposition:
Contrapositive of p → q is￢q →￢p and are equivalent.
Begin by assuming that￢q is true, i.e. q is false and proceed to prove that￢p is true, i.e. p is false. This will prove that￢q →￢p
is true. Because of equivalence, conclude that p → q is true.
Proof by Contradiction:
Start by assuming that P is false and arrive at a contradiction.
Contradiction may be in form that goes against some of the assumptions or some absurd conclusion.
The logic is as follows: We have prove that￢P → C is true. But C, being a contradiction, is false. Thus￢P is false and hence P is
true.
When P is of the form: ∀ x(P(x) → Q(x)), we proceed as follows:
We assume that the negation is true, i.e.∃ x￢(P(x) → Q(x)) or equivalently, ∃ x P(x)∧ ￢Q(x). So we start by assuming there is an
x for which P(x) is true and Q(x) is false.
Eventually, arrive at a contradiction, usually (but not always), in the form that P(x) is false.