Discrete mathematics_Homework_5_23_Pro_3

Part 1

Set P P is a well-formed formulas in propositional logic, involving T,F,s and {¬→∧∨↔} { ¬ → ∧ ∨ ↔ }

Basis Step:

T,F,s∈P T , F , s ∈ P , where s s is a propositional variable, are well-formed formulas.

Recursive Step:

If formulas A,B∈P A , B ∈ P , we have
(¬A)∈P,(A∧B)∈P,(A∨B)∈P,(A→B)∈P,(A↔B)∈P ( ¬ A ) ∈ P , ( A ∧ B ) ∈ P , ( A ∨ B ) ∈ P , ( A → B ) ∈ P , ( A ↔ B ) ∈ P

Part 2

f(x):P→N f ( x ) : P → N
f(x)=s f ( x ) = s , s s is the number of connectives for x,x∈P x , x ∈ P

Part 3

For variables p,q p , q , we have

p p	q q	¬(p∧q) ¬ ( p ∧ q )	¬p∨¬q ¬ p ∨ ¬ q
1	1	0	0
1	0	1	1
0	1	1	1
0	0	1	1

p p	q q	¬(p∨q) ¬ ( p ∨ q )	¬p∧¬q ¬ p ∧ ¬ q
1	1	0	0
1	0	0	0
0	1	0	0
0	0	1	1

p,q∈P p , q ∈ P , from the recursive definition, we know ∀A,B∈P ∀ A , B ∈ P , we holds De Morgan low.

Part 4

Firstly, ∗p=¬p ∗ p = ¬ p

p p	q q	¬p ¬ p	¬q ¬ q
1	1	0	0
1	0	0	1
0	1	1	0
0	0	1	1

From the tabel above, we know that If p=q p = q , ∗p=∗q ∗ p = ∗ q
p,q∈P p , q ∈ P , from the recursive definition, we know ∀A,B∈P ∀ A , B ∈ P , we holds duality principle.
IfA=B,∗A=∗B I f A = B , ∗ A = ∗ B