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# slidesfall2009 - Lecture Notes on DISCRETE MATHEMATICS...

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Lecture Notes on DISCRETE MATHEMATICS Eusebius Doedel 1

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LOGIC Introduction. First we introduce some basic concepts needed in our discussion of logic. These will be covered in more detail later. A set is a collection of “objects” (or “elements”). Examples. the infinite set of all integers : Z ≡ {· · · , 2 , 1 , 0 , 1 , 2 , 3 , · · ·} . the infinite set of all positive integers : Z + ≡ { 1 , 2 , 3 , · · ·} . the infinite set R of all real numbers. the finite set { T, F } , where T denotes “True” and F “False”. the finite set of alphabetic characters : { a, b, c, · · · , z } . the infinite set P n of all polynomial functions p ( x ) of degree n or less with integer coefficients. 2
A function (or “map”, or “operator”) is a rule that associates to every element of a set one element in another set. Examples. If S 1 = { a, b, c } and S 2 = { 1 , 2 } then the associations a mapsto→ 2 , b mapsto→ 1 , c mapsto→ 2 , define a function f from S 1 to S 2 . We write f : S 1 −→ S 2 . Similarly f ( n ) n 2 defines a function f : Z + −→ Z + . f ( n ) n 2 can also be viewed as a function f : Z −→ Z . 3

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The derivative operator D restricted to elements of P n can be viewed as a function from P n to P n - 1 , D : P n −→ P n - 1 , D : p mapsto→ dp dx . For example, if p ( x ) x 3 + 2 x , then D : x 3 + 2 x mapsto→ 3 x 2 + 2 , i.e. , D ( x 3 + 2 x ) 3 x 2 + 2 . 4
We can also define functions of several “variables”, e.g. , f ( x, y ) x + y , can be viewed as a function “from Z + cross Z + into Z + ”. We write f : Z + × Z + −→ Z + . 5

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Basic logical operators. The basic logical operators (“and” , “conjunction”) (“or” , “disjunction”) ¬ (“not” , “negation”) are defined in the tables below : p ¬ p T F F T p q p q T T T T F T F T T F F F p q p q T T T T F F F T F F F F 6
Let B ≡ { T, F } . Then we can view ¬ , , and , as functions ¬ : B −→ B, : B × B −→ B, : B × B −→ B. We can also view the arithmetic operators , +, and , as functions : Z −→ Z, + : Z × Z −→ Z, : Z × Z −→ Z, defined by value tables, for example, x x · · -2 2 -1 1 0 0 1 -1 2 -2 · · 7

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Logical expressions. A logical expression (or “proposition”) P ( p, q, · · · ) is a function P : B × B × · · · × B −→ B . For example, P 1 ( p, q ) p ∨ ¬ q and P 2 ( p, q, r ) p ( q r ) are logical expressions. Here P 1 : B × B −→ B , and P 2 : B × B × B −→ B . 8
The values of a logical expression can be listed in a truth table. Example. p q ¬ q p ( ¬ q ) T T F T T F T T F T F F F F T T 9

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Analogously, arithmetic expressions such as A 1 ( x, y ) x + ( y ) and A 2 ( x, y, z ) x ( y + z ) can be considered as functions A 1 : R × R −→ R , and A 2 : R × R × R −→ R , or, equivalently, A 1 : R 2 −→ R , and A 2 : R 3 −→ R . 10
Two propositions are equivalent if they always have the same values.

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slidesfall2009 - Lecture Notes on DISCRETE MATHEMATICS...

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