Discrete Mathematics
1.7.30
(i)If m^2=n^2, then m^2-n^2=(m+n)(m-n)=0. When the product of two
number is 0,at least one of them is 0,so m+n=0 or m-n=0,which means
m=n or m=-n
(ii)if m=n, then m^2=n^2. On the other hand, if m=-n, then m^2=(-1.6.28
Step/Reason
1. ∀x(P(x)∨Q(x))/Premise
2. ∀x((¬P(x) ∧ Q(x))→R(x))/Premise
3. P(c)∨Q(c)/Universal instantiation from (1)
4. (¬P(c) ∧ Q(c))→R(c) /Universal instantiation from (2)
5. ¬(¬P(c) ∧ Q(c))∨R(c)/Logical equivalence from (4)
6. (¬¬P(c) ∨¬Q(c) )∨R(c)/De Morgan’s law from (5)
7. (P(c) ∨¬Q(c)) ∨R(c)/Double negation law from (6)
8. (¬Q(c) ∨P(c)) ∨R(c)/Commutative law from (7)
9. ¬Q(c) ∨(P(c)∨R(c))/Associative law from (8)
10. P(c) ∨(P(c)∨R(c))/Resolution from (3) and (9)
11. (P(c) ∨P(c)) ∨R(c)/ Associative law from (10)
12.P(c) ∨R(c)/Idempotent law from (11)
13.R(c) ∨P(c)/Commutative law from (12)
14. ¬R(c) →P(c)/ Logical equivalence from (13)
15. ∀x(¬R(x)→P(x))/Universal generalization from (14)
16. (¬Q(c) ∨P(c)) ∨R(c)/Commutative law from (7)
17. ¬Q(c) ∨(P(c)∨R(c))/Associative law from (8)
18. P(c) ∨(P(c)∨R(c))/Resolution from (3) and (9)
19. (P(c) ∨P(c)) ∨R(c)/ Associative law from (10)
20.P(c) ∨R(c)/Idempotent law from (11)
21.R(c) ∨P(c)/Commutative law from (12)
22. ¬R(c) →P(c)/ Logical equivalence from (13)
23. ∀x(¬R(x)→P(x))/Universal generalization from (14)