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lect03c
rules of inference
Statements are different from proofs;
a proof is a *series* of statements, each with a justification
(involving previous statements, and rules)
Example:
Prove that
a ∨ (a∧b) and a
are equivalent: Proof:
| # | statement | justification |
|---|
| 1 |
a ∨ (a∧b) |
≡ |
(a∧T) ∨ (a∧b) |
T is identity for ∧ |
| 2 |
|
≡ |
a∧(T∨b) |
distribute ∧ over ∨ (in reverse) |
| 3 |
|
≡ |
a∧T |
dominance of T over ∨ |
| 4 |
|
≡ |
a |
T is identity for ∧ |
Alternate proof, using truth-tables:
| # | statement | justification |
|---|
| 1 |
the truth table for a∨(a∧b) is
| a | b | (a∧b) | a∨(a∧b) |
|---|
| F | F | F | F |
| F | T | F | F |
| T | F | F | T |
| T | T | T | T |
|
def'n of truth table |
| 2 |
the truth table for a is
|
def'n of truth table |
| 3 |
The column for a and for a∨(a∧b) is the same |
by inspection, and steps 1,2 |
| 4 |
Therefore a∨(a∧b) ≡ a |
by def'n of equivalent formulas, and step 3 |
Today we'll talk about a third type of proof: inference rules.
This is not about equivalence;
it's about inferring that one statement follows from another.
We can use it for things like:
(a ∧ b) ⇒ (a ∨ b)
(Note how these two statements are not equivalent1,
but one does imply the other.)
Rosen:
Look at inference rules, p.66 Table 1.
Go through exercises #3, #9a,b,c (conclusions only),
#7, #9b,c (using rules).
End with considering #9a, and the need
for inference-rules-for-quantifiers, Table 2.
1
If you're clever though, we can still use equivalence-type-proofs:
To prove φ⇒ψ, we show that
the single formula φ→ψ is a tautology (i.e., equivalent to T).
Note how
“↔” and “→” are
connectives within a single formula.
while
“≡” and “⇒” are
relations between formulas.
(In fact, ⇔ and ≡ both mean the same thing.)
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