CS250: Discrete Math for Computer Science. L5: PropCalc: Conditional Statements
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1 CS250: Discrete Math for Computer Science L5: PropCalc: Conditional Statements
2 p q p implies q if p then q p q Only the truth values of p and q matter, not the presence or absence of a causal relation between them. p q p q (p q) W p q p q p q (p q) (q p) W W W W
3 p q p implies q if p then q p q Only the truth values of p and q matter, not the presence or absence of a causal relation between them. p q q p contrapositve is equivalent. q p p q p q (p q) W p q p q p q (p q) (q p) W W W W
4 p q p implies q if p then q p q Only the truth values of p and q matter, not the presence or absence of a causal relation between them. p q q p contrapositve is equivalent. p q q p converse not equivalent. q p q p p q p q (p q) W p q p q p q (p q) (q p) W W W W
5 p q p implies q if p then q p q Only the truth values of p and q matter, not the presence or absence of a causal relation between them. p q q p contrapositve is equivalent. p q q p converse not equivalent. p q p q inverse not equivalent. q p q p p q p q p q (p q) W p q p q p q (p q) (q p) W W W W
6 p q p implies q if p then q p q Only the truth values of p and q matter, not the presence or absence of a causal relation between them. p q q p contrapositve is equivalent. p q q p converse not equivalent. p q p q inverse not equivalent. q p p q inverse is contrapositive of converse. q p q p p q p q p q (p q) W p q p q p q (p q) (q p) W W W W
7 English is ambiguous; PropCalc is precise. Translating between them can be subtle. p implies q q implies p if p then q if q then p p unless q p only if q p if q p iff q p is p is p is necessary sufficient necessary and sufficient for q for q for q p q p q q p p q q p
8 Natural Deduction R6: Our PropCalc proof rules are slightly different from Epp s proof rules. Important for the R6 quiz. F introduction p q p q p q p q p q p q p q p p F p p elimination p q p p q q p q p r q r r p q p p q q q p p F p p F p p p
9 Natural Deduction rule: -introduction 1 p 2 q 3 p q -i, 1, 2
10 Natural Deduction rule: -introduction 1 p 2 q 3 p q -i, 1, 2 Notation: W = a means that a is true in world, W.
11 Natural Deduction rule: -introduction 1 p 2 q 3 p q -i, 1, 2 Notation: W = a means that a is true in world, W. Proposition: -i is sound, i.e., if W = p and W = q then W = p q.
12 Natural Deduction rule: -introduction 1 p 2 q 3 p q -i, 1, 2 Notation: W = a means that a is true in world, W. Proposition: -i is sound, i.e., if W = p and W = q then W = p q. Proof. By definition of.
13 Natural Deduction rule: -elimination 1 p q 2 p -e, 1 3 q -e, 1
14 Natural Deduction rule: -elimination 1 p q 2 p -e, 1 3 q -e, 1 Proposition: -e is sound, i.e., if W = p q then W = p and W = q.
15 Natural Deduction rule: -elimination 1 p q 2 p -e, 1 3 q -e, 1 Proposition: -e is sound, i.e., if W = p q then W = p and W = q. Proof. By definition of.
16 Natural Deduction rule: -introduction 1 p 2 p q -i, 1 3 q p -i, 1
17 Natural Deduction rule: -introduction 1 p 2 p q -i, 1 3 q p -i, 1 Proposition: -i is sound, i.e., if W = p then W = p q and W = q p.
18 Natural Deduction rule: -introduction 1 p 2 p q -i, 1 3 q p -i, 1 Proposition: -i is sound, i.e., if W = p then W = p q and W = q p. Proof. By definition of.
19 Natural Deduction rule: -elimination 1 p q 2 p 3 q -e, 1, 2
20 Natural Deduction rule: -elimination 1 p q 2 p 3 q -e, 1, 2 Proposition: -e is sound, i.e., if W = p q and W = p then W = q.
21 Natural Deduction rule: -elimination 1 p q 2 p 3 q -e, 1, 2 Proposition: -e is sound, i.e., if W = p q and W = p then W = q. Proof. By definition of.
22 Natural Deduction rule: -elimination 1 p q 2 q 3 p -e, 1, 2
23 Natural Deduction rule: -elimination 1 p q 2 q 3 p -e, 1, 2 Proposition: -e is sound, i.e., if W = p q and W = q then W = p.
24 Natural Deduction rule: -elimination 1 p q 2 q 3 p -e, 1, 2 Proposition: -e is sound, i.e., if W = p q and W = q then W = p. Proof. If W = p q, then W = q p, the contrapositive. Then, by definition of, since W = q, we know that W = p
25 Natural Deduction rule: F-introduction 1 p 2 p 3 F F-i, 1, 2
26 Natural Deduction rule: F-introduction 1 p 2 p 3 F F-i, 1, 2 Proposition: F-i is sound, i.e., if W = p and W = p then W = F.
27 Natural Deduction rule: F-introduction 1 p 2 p 3 F F-i, 1, 2 Proposition: F-i is sound, i.e., if W = p and W = p then W = F. Proof. By definition of, if W = p, then W = p. Thus, it will never be the case that W = p and W = p. Thus, this proposition is vacuously true.
28 Natural Deduction rule: -introduction 1 p 2 p -i, 1
29 Natural Deduction rule: -introduction 1 p 2 p -i, 1 Proposition: -i is sound, i.e., if W = p then W = p.
30 Natural Deduction rule: -introduction 1 p 2 p -i, 1 Proposition: -i is sound, i.e., if W = p then W = p. Proof. p p
31 Natural Deduction rule: -elimination 1 p 2 p -e, 1
32 Natural Deduction rule: -elimination 1 p 2 p -e, 1 Proposition: -e is sound, i.e., if W = p then W = p.
33 Natural Deduction rule: -elimination 1 p 2 p -e, 1 Proposition: -e is sound, i.e., if W = p then W = p. Proof. p p
34 Natural Deduction R6: Our PropCalc proof rules are slightly different from Epp s proof rules. Important for the R6 quiz. F introduction p q p q p q p q p q p q p q p p F p p elimination p q p p q q p q p r q r r p q p p q q q p p F p p F p p p
35 R5 Quiz Answers 1. What is the contrapositive of p q? q p 2. What is the converse of p q? q p 3. What is the inverse of p q? p q Do the following equivlalences hold? 4. p q q p no 5. p q q p yes 6. q p q p no 7. q p p q yes 8. p q p q yes 9. p q p q yes 10. p q (p q) yes
36 Match the following English Statements with their meaning in PropCalc. 11. p if q q p 12. p only if q p q 13. p if and only if q p q 14. p unless q q p 15. r is a necessary condition for s to hold s r 16. r is a sufficient condition for s to hold r s 17. For s to hold, it is necessary and sufficient that r holds r s
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