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1 Discrete Structures Lecture 12 Implication III Read: Ch 4.1

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2 Additional Theorems re: (4.1) p (q p) (4.2) Monotonicity of V: proof in text (p q) (p V r q V r) (4.3) Monotonicity of : proof in class (p q) (p r q r)

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3 Abbreviation for Proving The transitivity of allows us to use an abbreviation for proofs involving implication. (3.82) Transitivity: (a) (p q) (q r) (p r) P Q R By transitivity (3.82 a), this proves that P R

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4 Combining and The mutual transitivity of and allows us to use a combination of Leibniz steps and the abbreviation using . (3.82) Transitivity: (b) (p q) (q r) (p r) P Q R By transitivity (3.82b), this proves that P R.

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5 Combining and The mutual transitivity of and allows us to use a combination of Leibniz steps and the abbreviation using . (3.82) Transitivity: (c) (p q) (q r) (p r) P Q R By transitivity (3.82c), this proves that P R.

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6 Using the Correct Hints When using this abbreviation, one must be careful to use the correct hints. Example of an incorrect hint: Assuming P1 Q1 is a theorem, we have x P1 x Q1 The correct hint SHOULD be: x P1 x Q1 Incorrect!

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7 Abbreviation for Proving cont. However, before this can be used, R must first be proved. Here is one proof of R (assuming P1 Q1 is a theorem). (P1 Q1) (x P1 x Q1) = true (x P1 x Q1) = x P1 x Q1

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8 Abbreviation for Proving cont. This is tortuous, so we can abbreviate as follows. x P1

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9 (4.3) Monotonicity of : (p q) (p r q r) Problem 4.2 says to prove (4.3) p r q r = ¬(p r) V (q r) = ¬p V ¬r V (q r) = (¬p V ¬r V q) (¬p V ¬r V r) = ¬p V ¬r V q ¬p V q = p q

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10 A Diversion on Abbreviation Mistakes There may be a tendency to try to write E[z:= P1] E[z:= Q1]

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11 Diversion Continued The step on the previous slide is not always sound. Here is an unsound use of this technique. ¬P1 ¬Q1 Here is a sound use. ¬P1 ¬Q1

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12 Metatheorem Parity This metatheorem explains exactly how a replacement of P by Q is to be made, where P Q.

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13 Metatheorem Parity Continued Metatheorem Parity. Consider a boolean expression E that: contains only operators ¬, V, and and has a single occurrence of a variable z. Suppose P Q is a theorem.

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14 Metatheorem Parity If z occurs within the scope of an even number of ¬ operations, then E[z:= P] E[z:= Q]. If z occurs within the scope of an odd number of ¬ operations, then E[z:= P] E[z:= Q].

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15 Metatheorem Parity For example, given P Q, Metatheorem Parity implies the following. R P R Q R ¬P R ¬Q

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16 Metatheorem Parity For example, given P Q, Metatheorem Parity implies the following. ¬(R P) ¬(R Q) ¬(R ¬P) ¬(R ¬Q)

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17 Metatheorem Parity For example, given P Q, Metatheorem Parity implies the following. ¬(R P) ¬(R Q) ¬R V ¬P ¬R V ¬Q ¬(R ¬P) ¬(R ¬Q) ¬R V ¬¬P ¬R V ¬¬Q Still even for Q’s scope. In P/Q’s scope, only one negative.

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18 Bonus Problem (4.4 in book) (p q) (r s) (p V r q V s) Problem 4.4 says to prove the above theorem (p q) (r s)

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