資訊科學數學 4 : More About Logic

1. 資訊科學數學4:More About Logic 陳光琦助理教授 (Kuang-Chi Chen) chichen6@mail.tcu.edu.tw

2. Logic - review Def. A proposition (p, q, r, …) is simply a statement (i.e., a declarative sentence) with a definite meaning, having a truth value that’s either true (T) or false (F) (never both, neither, or somewhere in between). • The unary negation operator “¬” (NOT) transforms a proposition into its logical negation. ¬p can be stated as “It’s not the case that p” ( ~p、~q(¬q)).

3. Basic Connectives& Truth Tables

4. Logical Connectives

5. Meaning of Implication • “q is implied by p” • “q follows from p” • “q if p” • “q when p” • “q whenever p” • “q is necessary for p” • “p implies q” • “if p, then q” • “if p, q” • “when p, q” • “whenever p, q” • “p only if q” • “p is sufficient for q”

6. Implication Implication： p→q，p 蘊含 q。 • “若 p則 q ” （if - then）。 • “p 對 q是充分的”or “p 是 q的充分條件”。 • “q 對 p是必要的”or “q 是 p的必要條件”。? • “p唯若 q ”。

7. Converse, Inverse, Contrapositive Some terminology, for an implication p  q: • Its converseis: q  p . (逆命題) • Its inverse is: ¬p  ¬q . (反逆命題) • Its contrapositive: ¬q  ¬p . (變換命題) • One of these three has the same meaning (same truth table) as p q. Can you get it? Contrapositive !

8. Some Alternative Notations

9. Bits & Bit Operations • A bit is a binary (base 2) digit: 0 or 1. • Bits may be used to represent truth values. • By convention: 0 represents “false”, 1 represents “true”. • Boolean algebra is like ordinary algebra except that variables stand for bits, + means “or”, and multiplication means “and”. • More details later.

10. Tautology & Contradiction • A tautology is a compound proposition that is trueno matter what the truth values of its atomic propositions are! Eg.,p  p. • A contradictionis a compound proposition that is false no matter what! Eg.,p  p. • Other compound propositions are contingencies.

11. Tautology & Contradiction Def. A compound statement is called a tautologyif it is true for all truth value assignments for its component statement. If a compound statement is false for all such assignments, then it is called a contradiction. • Throughout this course we will use T0 and F0 to represent any tautology and contradiction, resp. # We use the ideas of tautology and implication to describe what we mean by a valid argument.

12. Tautology & Contradiction • 若複合敘述對其敘述的所有真假值均為真，則被稱為重言（tautology，T0）；若複合敘述對其敘述的所有真假值均為假，則被稱為矛盾（contradiction，F0 ）。 # 在資訊科學，使用 重言 及 蘊含 的概念來描述一個 “有效的論證（valid argument）”。

13. Logical Equivalence:The Laws of Logic

14. Logical Equivalence • Compound proposition p is logically equivalentto compound proposition q, written p≡ q, IFFthe compound proposition pqis a tautology. • Compound propositions p and q are logically equivalent to each otherIFFp and q contain the same truth values as each other in all rows of their truth tables.

15. Logically Equivalent Logically Equivalent (⇔, ≡) • 邏輯相等、實質相同、邏輯等價 • 兩敘述若邏輯等價 (logically equivalent)，s1⇔s2，意謂 “當 敘述 s1為真時，若且唯若 敘述 s2為真”，“當 敘述 s1為假時，若且唯若 敘述 s2為假”。 • E.g., (p→q) ⇔ (~pⅴq) (p↔q) ⇔ (p→q) ^ (q→p) (p↔q) ⇔ (~pⅴq) ^ (~qⅴp)

16. Example E.g., (p→q) ⇔ (~pⅴq) (p↔q) ⇔ (p→q) ^ (q→p) (p↔q) ⇔ (~pⅴq) ^ (~qⅴp)

17. Equivalence Law • These are similar to the arithmetic identities you may have learned in algebra, but for propositional equivalences instead. • They provide a pattern or template that can be used to match all or part of a much more complicated proposition and to find an equivalence for it.

18. Logic Law (p.1)

19. Logic Law (p.2)

20. Example 1 • In Pascal program segments shown below, x, y, z, and i are integer variables.

21. Example 1 (cont’d) Part (a) z : = 4 ; For i : = 1 to 10 do Begin x : = z – i ; y : = z + 3*i ; If (( x > 0 ) and ( y > 0 )) then Writeln (‘ The value of the sum x + y is ’, x + y ) End;

22. Example 1 (cont’d) Part (b) z : = 4 ; For i : = 1 to 10 do Begin x : = z – i ; y : = z + 3*i ; If x > 0 then If y > 0 then Writeln (‘ The value of the sum x + y is ’, x + y ) End;

23. Example 1 (cont’d) • Part (a) use a decision structure comparable to a statement of the form (p ^ q)→r. • In part (b), we have p: x>0, q: y>0, which are statements when the variables x, y are assigned the values 4 – i (for x) and 4 + 3*i (for y). • Now, the letter r denotes the Writeln statement, an “executable statement” that is actually not a statement in the usual sense of a declarative sentence, which can be labeled true or false.

24. Example 1 (cont’d) • In part (a), the total number of comparisons, (x > 0) and (y > 0), is 10 (for x > 0) + 10 (for y > 0) = 20. • In part (b), it’s the nested implications p→ (q→ r), therefore, (y > 0) is not executed unless the comparison (x > 0) is executed and evaluated as true. The total number of comparisons is 10 (for x > 0) + 3 (for y > 0, as i = 1, 2, 3) = 13. • Hence, program (b) is more efficient than (a).

25. Example 2 Find a simpler way of the compound statement. Hence, we see that (pⅴq)^ ~(~p ^ q) ⇔ p .

26. Example 3 • A switching network is made up of wires and switches connecting two terminals T1 and T2. In such a network, and switch is either open (0), or closed (1). • We have three networks. Part (a) contains one switch, and each of parts (b) and (c) contains two (independent) switches.

27. Example 3 (cont’d) • Part (a): Part (b): Part (c):

28. Example 3 (cont’d) • For part (a), a network with one switch. • For part (b), current flows from T1to T2 if either of the switches p or q is closed. We call this a parallel network and represent it by pⅴq. • For part (c), it requires that each of the switches p, q be closed in order for current to flow from T1 to T2. Here the switches are in series, and this network is represented by p ^ q.

29. Example 4 • The switches in a network need not act independently of each other.

30. Example 4 (cont’d) • The network is represented the corresponding statement (pⅴqⅴr)^(pⅴtⅴ~q)^(pⅴ~tⅴr). Using the laws of logic, we can find a simpler statement.

31. Example 4 (cont’d)

32. Example 4 (cont’d) • Hence (pⅴqⅴr)^(pⅴtⅴ~q)^(pⅴ~tⅴr) ⇔ pⅴ[(tⅴ~q) ^ r], and the new network shown below, which is equivalent to the original one. • This one has only four switches, five fewer than the original one.

33. Defining Operators via Equivalences Using equivalences, we can define operators in terms of other operators. • Exclusive or : pq≡ (pq)(pq)pq≡ (pq)(qp) • Implies : pq ≡ p  q • Biconditional : pq ≡ (pq) (qp)pq ≡ (pq)

34. Summary of Propositional Logic • Atomic propositions: p, q, r, … • Boolean operators:       • Compound propositions: s : (p q)  r • Equivalences:p q  (p  q) • Proof of equivalence: • Truth tables. • Symbolic derivations. p q  r …

35. Logical Implication:Rules of Inference

36. A Valid Argument Helping in proving theorems through the texts. • The general form of an argument: e.g., the implication: (p1 p2 p3 … pn)→ q . • Here n is a positive integer, the statements p1, p2, p3, …,pn are called the premises of the argument, and the statement q is conclusion for the argument. • The argument is valid if whenever : • Each premise p1, p2, p3, ..., pn is true, • then the conclusion q is likewise true.

37. A Valid Argument (cont’d) # In fact, if any one of p1, p2, p3, ..., pn is false, then the hypothesis p1 p2 p3 … pn is false and the implication (p1 p2 p3 … pn)→ qis automatically TRUE, regardless of the truth value of q. • Hence, one way to establish the validity of a given argument is to show that the statements (p1 p2 p3 … pn)→ qis a TAUTOLOGY. (all truths)

38. Example Let p, q, r be the primitive statements given as • p: Roger studies; q: Roger plays tennis; r: Roger passes discrete mathematics. Now, let p1,p2, p3 denote the premises • p1: If Roger studies, then he will pass discrete math. • p2: If Roger doesn’t play tennis, then he’ll study. • p3: Roger failed discrete math. Q: We want to determine whether the argument (p1 p2 p3)→ q is valid.

39. Example (cont’d) To do so, we rewrite p1,p2, p3 as • p1: p→r ; • p2: ~q→p ; • p3: ~r . and examine the truth table for the implication (p1 p2 p3) → q i.e.,[(p→r)  (~q→p )  ~r] → q .

40. Example (cont’d) Since the final column contains all 1’s, the implication is a tautology. Hence (p1 p2 p3)→ q is a valid argument.

41. Logically Implication Def. If p, q are arbitrary statements such that p→q is a tautology, then we say that p logically implies q and we write p⇒q to denote this situation. • When p, q are any statements and p⇒q, the implication p→q is a tautology, and we refer to p →q as a logical implication. # We can avoid dealing with the idea of a tautology here by saying that p⇒q (i.e., p logically implies q) if q is true whenever p is true.

42. Logically Implication (cont’d) Let p, q be arbitrary statements. 1) If p⇔q, then the statement p↔q is a tautology, so the statements have the same truth values. Under the conditions the statements p→q, q→p are tautologies, and we have p⇒q and q⇒p. 2) Conversely, suppose that p⇒q and q⇒p. The logical implicationp→qtells us that we never have statement p with the truth value 1 and q with the truth value 0. But could we have q with the truth value 1 and p with the truth value 0 ?

43. Logically Implication (cont’d) • If this occurred, we could not have the logical implication q→p. Therefore, when p⇒q and q⇒p, the statements p, q have the same truth values and p⇔q. • Finally, the notation p⇒q is used to indicate that p→q is not a tautology – so the given implication (p→q) is not a logical implication.