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How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Outline
1 Recap
2 the form P ∧ QGoals of the form P ∧ Q
3 the form P ↔ QTo prove a goal of the form P ↔ Q
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form P → Q
Assume P is true and then prove Q
Before using strategy
Givens-
GoalP → Q
After using strategy
Givens--P
GoalQ
Form of final proof
Suppose P.[Proof of Q goes here].
Therefore, P → Q.
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form P → Q
Assume P is true and then prove Q
Before using strategy
Givens-
GoalP → Q
After using strategy
Givens--P
GoalQ
Form of final proof
Suppose P.[Proof of Q goes here].
Therefore, P → Q.
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form P → Q
Assume P is true and then prove Q
Before using strategy
Givens-
GoalP → Q
After using strategy
Givens--P
GoalQ
Form of final proof
Suppose P.[Proof of Q goes here].
Therefore, P → Q.
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form P → Q
Assume P is true and then prove Q
Before using strategy
Givens-
GoalP → Q
After using strategy
Givens--¬Q
Goal¬P
Form of final proof
Suppose Q is false.[Proof of ¬P goes here].
Therefore, P → Q.
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form P → Q
Assume P is true and then prove Q
Before using strategy
Givens-
GoalP → Q
After using strategy
Givens--¬Q
Goal¬P
Form of final proof
Suppose Q is false.[Proof of ¬P goes here].
Therefore, P → Q.
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form P → Q
Assume P is true and then prove Q
Before using strategy
Givens-
GoalP → Q
After using strategy
Givens--¬Q
Goal¬P
Form of final proof
Suppose Q is false.[Proof of ¬P goes here].
Therefore, P → Q.
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form ¬P
If possible, re-express the goal in another form and use aprevious strategy for the re-expressed goal.
Some useful laws to transform a goal
Associative laws Contrapositive law Double negation lawCommutative laws DeMorgan’s laws Idempotent lawsConditional laws Distributive laws Quantifier negation laws
Conditional laws
P → Q is equivalent to ¬P ∨ QP → Q is equivalent to ¬(P ∧ ¬Q)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form ¬P
If possible, re-express the goal in another form and use aprevious strategy for the re-expressed goal.
Some useful laws to transform a goal
Associative laws Contrapositive law Double negation lawCommutative laws DeMorgan’s laws Idempotent lawsConditional laws Distributive laws Quantifier negation laws
Conditional laws
P → Q is equivalent to ¬P ∨ QP → Q is equivalent to ¬(P ∧ ¬Q)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form ¬P
Assume P is true and try to reach a contradiction. Once youhave reached a contradiction, you can conclude that P must befalse.
Before using strategy
Givens-
Goal¬P
After using strategy
Givens-P
GoalContradiction
Form of final proof
Suppose P is true.Proof of contradiction goes here.
Thus, P is false.
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form ¬P
Assume P is true and try to reach a contradiction. Once youhave reached a contradiction, you can conclude that P must befalse.
Before using strategy
Givens-
Goal¬P
After using strategy
Givens-P
GoalContradiction
Form of final proof
Suppose P is true.Proof of contradiction goes here.
Thus, P is false.
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form ¬P
Assume P is true and try to reach a contradiction. Once youhave reached a contradiction, you can conclude that P must befalse.
Before using strategy
Givens-
Goal¬P
After using strategy
Givens-P
GoalContradiction
Form of final proof
Suppose P is true.Proof of contradiction goes here.
Thus, P is false.
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To use a given of the form ¬P
If you are doing a proof by contraction, try making P yourgoal. if you can prove P, then the proof will be complete,because P contradicts the given ¬P.
Before using strategy
Givens¬P-
GoalContradiction
After using strategy
Givens¬P-
GoalP
Form of final proof
Proof of P goes here.Since we already know ¬P, this is a contradiction.
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To use a given of the form ¬P
If you are doing a proof by contraction, try making P yourgoal. if you can prove P, then the proof will be complete,because P contradicts the given ¬P.
Before using strategy
Givens¬P-
GoalContradiction
After using strategy
Givens¬P-
GoalP
Form of final proof
Proof of P goes here.Since we already know ¬P, this is a contradiction.
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To use a given of the form ¬P
If you are doing a proof by contraction, try making P yourgoal. if you can prove P, then the proof will be complete,because P contradicts the given ¬P.
Before using strategy
Givens¬P-
GoalContradiction
After using strategy
Givens¬P-
GoalP
Form of final proof
Proof of P goes here.Since we already know ¬P, this is a contradiction.
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To use a given of the form ¬P
If possible re-express this given in some other form.
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
Modus ponens
If we have givensP → Q andPthen we can concludeQ is true.
Modus tollens
If we have givensP → Q and¬Qthen we can conclude¬P is true.
Some advice
Modus ponens and modus tollens can be very useful in proofsand should probably be applied immediately wheneverapplicable in a proof
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
Modus ponens
If we have givensP → Q andPthen we can concludeQ is true.
Modus tollens
If we have givensP → Q and¬Qthen we can conclude¬P is true.
Some advice
Modus ponens and modus tollens can be very useful in proofsand should probably be applied immediately wheneverapplicable in a proof
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form ∀xP(x)
Let x stand for an arbitrary object and prove P(x).
Before using strategy
Givens-
Goal∀xP(x)
After using strategy
Givens--
GoalP(x)
Form of final proof
Let x be arbitrary.[Proof of P(x) goes here].
Since x was arbitrary, we can conclude that ∀xP(x)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form ∀xP(x)
Let x stand for an arbitrary object and prove P(x).
Before using strategy
Givens-
Goal∀xP(x)
After using strategy
Givens--
GoalP(x)
Form of final proof
Let x be arbitrary.[Proof of P(x) goes here].
Since x was arbitrary, we can conclude that ∀xP(x)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form ∀xP(x)
Let x stand for an arbitrary object and prove P(x).
Before using strategy
Givens-
Goal∀xP(x)
After using strategy
Givens--
GoalP(x)
Form of final proof
Let x be arbitrary.[Proof of P(x) goes here].
Since x was arbitrary, we can conclude that ∀xP(x)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form ∃xP(x)
Try to find a value of x for which you think P(x) will be true.
Before using strategy
Givens--
Goal∃xP(x)
After using strategy
Givens---
GoalP(x)
x= (the value you decided on)
Form of final proof
Let x= (the value you decided on).[Proof of P(x) goes here].
Thus ∃xP(x)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form ∃xP(x)
Try to find a value of x for which you think P(x) will be true.
Before using strategy
Givens--
Goal∃xP(x)
After using strategy
Givens---
GoalP(x)
x= (the value you decided on)
Form of final proof
Let x= (the value you decided on).[Proof of P(x) goes here].
Thus ∃xP(x)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form ∃xP(x)
Try to find a value of x for which you think P(x) will be true.
Before using strategy
Givens--
Goal∃xP(x)
After using strategy
Givens---
GoalP(x)
x= (the value you decided on)
Form of final proof
Let x= (the value you decided on).[Proof of P(x) goes here].
Thus ∃xP(x)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a given of the form ∃xP(x)
Introduce a new variable x0 into the proof to stand for anobject for which P(x0) is true.
Before using strategy
Givens∃xP(x)-
Goal-
After using strategy
GivensP(x0)--
Goal−
Form of final proof
Let P(x0).
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a given of the form ∃xP(x)
Introduce a new variable x0 into the proof to stand for anobject for which P(x0) is true.
Before using strategy
Givens∃xP(x)-
Goal-
After using strategy
GivensP(x0)--
Goal−
Form of final proof
Let P(x0).
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a given of the form ∃xP(x)
Introduce a new variable x0 into the proof to stand for anobject for which P(x0) is true.
Before using strategy
Givens∃xP(x)-
Goal-
After using strategy
GivensP(x0)--
Goal−
Form of final proof
Let P(x0).
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a given of the form ∀xP(x)
You can plug in any value say a for x and use this to concludethat P(a) is true.
Before using strategy
Givens∀xP(x)
Goal-
After using strategy
GivensP(a)-
Goal−
Useful if P(a) or ¬P(a) appears in the proof somewhere.
Form of final proof
Since p(a),
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a given of the form ∀xP(x)
You can plug in any value say a for x and use this to concludethat P(a) is true.
Before using strategy
Givens∀xP(x)
Goal-
After using strategy
GivensP(a)-
Goal−
Useful if P(a) or ¬P(a) appears in the proof somewhere.
Form of final proof
Since p(a),
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a given of the form ∀xP(x)
You can plug in any value say a for x and use this to concludethat P(a) is true.
Before using strategy
Givens∀xP(x)
Goal-
After using strategy
GivensP(a)-
Goal−
Useful if P(a) or ¬P(a) appears in the proof somewhere.
Form of final proof
Since p(a),
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form P ∧ Q
Prove P and Q seperately
givens of the form P ∧ Q
Treat this as two separate givens: P and Q seperately
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose A ⊆ B, and A and B are disjoint. Prove thatA ⊆ B \ C .
GivensA ⊆ BA ∩ C = ∅x ∈ A
Goalx ∈ B
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose A ⊆ B, and A and B are disjoint. Prove thatA ⊆ B \ C .
GivensA ⊆ BA ∩ C = ∅
x ∈ A
GoalA ⊆ B \ C
x ∈ B
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose A ⊆ B, and A and B are disjoint. Prove thatA ⊆ B \ C .
GivensA ⊆ BA ∩ C = ∅
x ∈ A
GoalA ⊆ B \ C
x ∈ B
A ⊆ B \ C is equivalent to ∀x(x ∈ A → x ∈ B \ C )
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose A ⊆ B, and A and B are disjoint. Prove thatA ⊆ B \ C .
GivensA ⊆ BA ∩ C = ∅
x ∈ A
Goalx ∈ A → x ∈ B\C
x ∈ B
x is arbitrary
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose A ⊆ B, and A and B are disjoint. Prove thatA ⊆ B \ C .
GivensA ⊆ BA ∩ C = ∅x ∈ A
Goalx ∈ B \ C
x ∈ B
x ∈ (B \ C ) is equivalent to x ∈ B ∧ x /∈ C
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose A ⊆ B, and A and B are disjoint. Prove thatA ⊆ B \ C .
GivensA ⊆ BA ∩ C = ∅x ∈ A
Goalx ∈ B∧ /∈ C
x ∈ B
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose A ⊆ B, and A and B are disjoint. Prove thatA ⊆ B \ C .
GivensA ⊆ BA ∩ C = ∅x ∈ A
Goalx ∈ Bx /∈ C
By apply the rule conjunction
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose A ⊆ B, and A and B are disjoint. Prove thatA ⊆ B \ C .
GivensA ⊆ BA ∩ C = ∅x ∈ A
Goal
x ∈ B
x /∈ C
By definition of ⊆
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose A ⊆ B, and A and B are disjoint. Prove thatA ⊆ B \ C .
GivensA ⊆ BA ∩ C = ∅x ∈ A
Goal
x ∈ B
By expaning A ∩ C = ∅
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose A ⊆ B, and A and B are disjoint. Prove thatA ⊆ B \ C .
GivensA ⊆ BA ∩ C = ∅x ∈ A
Goal
x ∈ B
Suppose x ∈ A. Since A ⊆ B, it follows that x ∈ B, and sinceA and C are disjoint, we must have x ∈ C . Thus, x ∈ B \ C .Since x was an arbitrary element of A, we can conclude thatA ⊆ B \ C
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Method
To prove a goal of the form P ↔ Q
Prove P → Q and Q → P seperately
givens of the form P ↔ Q
Treat this as two separate givens: P → Q and Q → Pseperately
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose x is an integer. Prove that x is even iff x2 is even
Givensx ∈ Zk ∈ Zx = 2× k
Goal
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose x is an integer. Prove that x is even iff x2 is even
Givensx ∈ Z
k ∈ Zx = 2× k
Goal
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose x is an integer. Prove that x is even iff x2 is even
Givensx ∈ Z
k ∈ Zx = 2× k
Goal∀ x (x is even P ↔ Q x2 is even)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose x is an integer. Prove that x is even iff x2 is even
Givensx ∈ Z
k ∈ Zx = 2× k
Goal(x is even P ↔ Q x2 is even)
Let x be arbitrary
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose x is an integer. Prove that x is even iff x2 is even
Givensx ∈ Z
k ∈ Zx = 2× k
Goal(x is even P → Q x2 is even)(x2 is even P → Q x is even)
Applying P ↔ Q rule
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose x is an integer. Prove that x is even iff x2 is even
Givensx ∈ Z
k ∈ Zx = 2× k
Goal(x is even P → Q x2 is even)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose x is an integer. Prove that x is even iff x2 is even
Givensx ∈ Zx is even
k ∈ Zx = 2× k
Goalx2 is even
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose x is an integer. Prove that x is even iff x2 is even
Givensx ∈ Z∃k ∈ Z(x = 2k)
k ∈ Zx = 2× k
Goal∃k ∈ Z(x2 = 2k)
Definition of even
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose x is an integer. Prove that x is even iff x2 is even
Givensx ∈ Zk ∈ Zx = 2× k
Goal∃k ∈ Z(x2 = 2k)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose x is an integer. Prove that x is even iff x2 is even
Givensx ∈ Zk ∈ Zx = 2× k
Goal(x2 = (2× k)2)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose x is an integer. Prove that x is even iff x2 is even
Givensx ∈ Zk ∈ Zx = 2× k
Goalx2 = 4× k2
Suppose x is even. Then for some integer k, x =2× k.Therefore, x2 = 4× k2, so since 22 is even. Thus, if x is eventhen x2 is even.Similarly other direction also holds
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
∀x¬P(x) ↔ ¬∃xP(x)
Givens¬∃xP(x)
Goal∃xP(x)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
∀x¬P(x) ↔ ¬∃xP(x)
Givens∀x¬P(x)
¬∃xP(x)
Goallnot∃xP(x)
∃xP(x)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
∀x¬P(x) ↔ ¬∃xP(x)
Givens∀x¬P(x)∃xP(x)
¬∃xP(x)
Goalcontradiction
∃xP(x)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
∀x¬P(x) ↔ ¬∃xP(x)
Givens∀x¬P(x) P(x)
¬∃xP(x)
Goalcontradiction
∃xP(x)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
∀x¬P(x) ↔ ¬∃xP(x)
Givens∀x¬P(x) P(x)
¬∃xP(x)
Goalcontradiction
∃xP(x)
Plugging x
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
∀x¬P(x) ↔ ¬∃xP(x)
Givens¬∃xP(x)
Goal∀x¬P(x)
∃xP(x)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
∀x¬P(x) ↔ ¬∃xP(x)
Givens¬∃xP(x)
Goal¬P(x)
∃xP(x)
x be an arbitrary
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
∀x¬P(x) ↔ ¬∃xP(x)
Givens¬∃xP(x)P(x)
GoalContradiction
∃xP(x)
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
∀x¬P(x) ↔ ¬∃xP(x)
Givens¬∃xP(x)P(x)
Goal∃xP(x)
Suppose lnot∃xP(x). Let x be arbitrary, and suppose P(x).Since we have a specific x for which P(x) is true, it follows that∃x .P(x), which is a contradiction. Therefore, 6 P(x).
How To ProveIt
D. Gift Samue
Recap
the formP ∧ Q
Goals of theform P ∧ Q
the formP ↔ Q
To prove a goalof the formP ↔ Q
Example
Suppose A,B, and C are sets. Then A ∪ (B \ C ) = (A ∩ B) \ C
For every integer, 6 | n iff 2 | n and 3 | n
