-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
-
Useful Equivalence:
-
Useful Equivalence:
-
Useful Equivalence:
-
-
-
-
-
-
-
-
-
Rule of Disjunctive Syllogism
-
Rule of Disjunctive Syllogism
-
-
-
Rule of Conjunctive Simplification
-
Rule of Conjunctive Simplification
-
Rule of Disjunctive Amplification
-
Rule of Disjunctive Amplification
-
Rule of Conditional Proof
-
Rule of Conditional Proof
-
-
-
Rule of the Constructive Dilema
-
Rule of the Constructive Dilema
-
Rule of the Destructive Dilemma
-
Rule of the Destructive Dilemma
-
The Rule of Universal Specification
- If an open statement becomes true for all replacements by the members in a given universe, then that open statement is true for each specific individual member in that universe.
-
The Rule of Universal Generalization
- If an open statement p(x) is proved to be true when x is replaced by an arbitrarily chosen element c from our universe, then the universally quantified statement for all x, p(x) is true.
-
Forward-Backward Method
When to use?
What to assume?
What to conclude?
How to use?
- When to use: As a first attempt
- What to assume: A
- What to conclude: B
- How to use: Work forward from A, and backward from B
-
Contrapositive Method
When to use?
What to assume?
What to conclude?
How to use?
- When to use: When B contains negation
- What to assume: NOT B
- What to conclude: NOT A
- How to use: Work forward from NOT B, and backward from NOT A
-
Contradiction Method
When to use?
What to assume?
What to conclude?
How to use?
- When to use: When B contains negation, or when Forward-Backward and Contrapositive methods fail.
- What to assume: A AND NOT B
- What to conclude: Some contradiction
- How to use: Word forward from A AND NOT B
-
Construction Method
When to use?
What to assume?
What to conclude?
How to use?
- When to use: When B contains existence
- What to assume: A
- What to conclude: That there is the desired object
- How to use: Construct the object, then show that it has the certain property and that something happens
-
Choose Method
When to use?
What to assume?
What to conclude?
How to use?
- When to use: When B contains forall
- What to assume: A, and choose an object with the certain property
- What to conclude: That something happens
- How to use: Work forward from A and the fact that the object has the certain property, also backwards from the something that happens
-
Specialization Method
When to use?
What to assume?
What to conclude?
How to use?
- When to use: When A contains forall
- What to assume: A
- What to conclude: B
- How to use: Work forward by specializing A to one particular object having the certain property
-
Forward Uniqueness Method
When to use?
What to assume?
What to conclude?
How to use?
- When to use: When A contains "unique"
- What to assume: That there is such an object, X
- What to conclude: That X and Y are the same, that is, X=Y
- How to use: Look for another object Y with the same properties as X
-
Direct Uniqueness Method
When to use?
What to assume?
What to conclude?
How to use?
- When to use: When B contains "unique"
- What to assume: That there are two such objects, and A
- What to conclude: The two objects are equal
- How to use: Work forward using A and the properties of the object, also work backward to shoe the objects are equal
-
Indirect Uniqueness Method
When to use?
What to assume?
What to conclude?
How to use?
- When to use: When B contains "unique"
- What to assume: There are two different objects, and A
- What to conclude: Some contradiction
- How to use: Work forward from A using the properties of the two objects and the fact that they are different
-
Induction Method
When to use?
What to assume?
What to conclude?
How to use?
- When to use: When a statement P(n) is true for each integer n >= n0
- What to assume: P(n) is true for n
- What to conclude: P(n0) is true, P(n+1) is true
- How to use: First prove P(n0), then use the assumption P(n) is true to prove that P(n+1) is true
-
Proof by Cases
When to use?
What to assume?
What to conclude?
How to use?
- When to use: When A has the form "C OR D"
- What to assume: Case 1: C / Case 2: D
- What to conclude: B
- How to use: First prove that C -> B, then prove D -> B
-
Proof by Elimination
When to use?
What to assume?
What to conclude?
How to use?
- When to use: When B has the form "C OR D"
- What to assume: (A AND NOT C) or (A AND NOT D)
- What to conclude: D or C
- How to use: Work forward from (A AND NOT C), and backward from D; or work forward from (A AND NOT D), and backward from C
-
Max/Min 1 Method
When to use?
What to assume?
What to conclude?
How to use?
- When to use: When A or B has the form "max S <= z" or "min S >= z"
- What to assume: (nothing)
- What to conclude: (nothing)
- How to use: Convert to "for all s in S, s <= z or s >= z", then choose (if in B) or specialization (if in A)
-
Max/Min 2 Method
When to use?
What to assume?
What to conclude?
How to use?
- When to use: When A or B has the form "max S >= z" or "min S <= z"
- What to assume: (nothing)
- What to conclude: (nothing)
- How to use: Convert to "there is an s in S such that s >= z or s <= z", then work forward (if in A) or use construction (if in B)
|
|
