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Text 2 Johann K. Brunner Mathematics, October 2010 Basics in Mathematical Logic 1 Assertions In mathematics we generally deal with assertions (or statements), which are sentences that are either true or wrong, like "The mathematics course is the …rst course in the Innsbruck-Linz PhD-program in Economics".1 :A (= nonA) is the negation of an assertion A. :A is true if A is wrong and wrong if A is true. By using the logical operators AND (^); OR (_), assertions can be combined to a new assertion: Let A and B be assertions, - then C = A ^ B is a new assertion, which is true if A and B are both true, otherwise it is wrong. - then D = A _ B is a new assertion, which is true if at least one of A and B is true, it is wrong if both A and B are wrong. Clearly, combined assertion like C or D can again be combined to more complex statements. A particularly important combination of two assertion A, B is A ) B; which is de…ned as :A _ B: A ) B should be read as "A implies B" or "If A then B". Note that if we know that the assertion A ) B is true and that A is true, it follows that B must be true. On the other hand, if we know that A ) B is true and that B is wrong, then A must be wrong as well (i. e., :A must be true). 1 Thus, we do not talk about phrases that express emotions ("What a wonderful day!") or requests ("Come with me!"). 1 In fact, the assertion A ) B is completely equivalent to the assertion :B ) :A (which, by de…nition means B _ :A; which clearly is true whenever :A _ B is true). If A ) B is true, we say that A is a su¢ cient condition for B, while B is a necessary condition for A. (Note the di¤erence!) Example: If a natural number is divisible by 6 (without rest), then it is divisible by 3 (without rest). Two assertions A; B are called equivalent, if both A ) B and B ) A hold, which is abbreviated by A , B: If A , B holds, A is true if and only if B is true (and, trivially, the other way round).2 Example: A natural number is divisible by 6 (without rest) if and only if it is divisible by 3 (without rest) and by 2 without rest. 2 Theorems and proofs An interesting mathematical statement (= assertion) is usually called a theorem or a proposition. A lemma is a statement which is itself not that important, but is useful to prove a theorem or a proposition. Statements typically have the form "A ) B"; where A is a - possibly combined - statement (the assumption(s) or condition(s)) and B is a possibly combined - statement (the implications(s) or consequence). Frequently, not all assumptions are mentioned explicitly, because it is understood that the statement is made within a given framework. To give an example: Theorem 1: If good i is a normal good, it cannot be a Gi¤en good. Here it is understood that we argue within the standard consumer theory, where households can consume n di¤erent goods, maximise utility and observe the budget constraint. In order to prove a statement A ) B; we transform A and B until we arrive at a formulation which is known to be true. That is, we try to …nd implications of A (or equivalent statements of A) which are known to imply B, or to imply a known su¢ cient 2 Instead of "if and only if" sometimes the abbreviation "i¤" is used: A , B means A is true i¤ B is true. 2 condition for B. Example: Proof of Theorem 1: Consider the demand function fi for good i, depending on prices p1 ; :::; pn and the budget b, fi (p1 ; :::; pn ; b): Good i is a normal good by assumption, which means that @fi @b 0: (1) By the Slutsky-equation, the (uncompensated) price-e¤ect can be separated into a substitution e¤ect and an income e¤ect: @f c @fi = i @pi @pi xi @fi ; @b (2) where fic denotes the compensated (or Hicksian) demand function and xi = fi (p1 ; :::; pn ; b) denotes the quantity of good i. We know from standard consumer theory that the own compensated price e¤ect is always negative: @fic < 0: @pi If we use (1) and (3) in (2), we …nd @fi @pi (3) < 0; which means that good i is not a Gi¤en good. 3 Note that equivalent ways to express the meaning of Theorem 1 would be: A normal good is never a Gi¤ en good. Let good i be a normal good. Then good i cannot be a Gi¤ en good. Moreover, as :B ) :A is equivalent to A ) B; Theorem 1 can also be written as: If good i is a Gi¤ en good, then it cannot be a normal good. In fact, a statement A ) B is sometimes proved by contradiction, i. e. by showing that if B is wrong it follows that A is also wrong (:A is implied by :B): A speci…c technique to prove a statement is called proof by induction. This technique 3 (or a proof. or the letters QED - "quod erat demonstrandum") is frequently used to designate the end of 3 can be used, if natural numbers are involved in the statement. It consists of two steps: Step 1: show that the statement is true for some (low) natural number n0 ; usually for 1. Step 2: show that if the statement is true for some arbitrary number n, then it is also true for the number n + 1: These two steps allow the conclusion that the statement is true for all natural numbers. To give an example: Theorem 2: Let n be a natural number. Then sum of the …rst n odd numbers is equal to n2 ; i. e.: 1 + 3 + 5 + ::: + (2n 1) = n2 : (*) Proof of Theorem 2: Note …rst that the sum on the left-hand-side (LHS) of (*) consists indeed of n successive odd numbers, starting with 1. Step 1: Let n = 1. Then the statement is true because 2n 1 = 1 (the sum in (*) has only 1 element), and 1 = 12 : Step 2: Assume that the statement is true for some arbitrary (but odd) number n: 1 + 3 + 5 + ::: + (2n 1) = n2 : Add the next odd number 2n + 1 on both sides to get 1 + 3 + 5 + ::: + (2n 1) + (2n + 1) = n2 + 2n + 1 Now we have the sum of the …rst n+1 odd numbers on the LHS. Clearly, the righthand side (RHS) is equal to (n + 1)2 : Thus the statement is true for n + 1; given that it is true for n. Conclusion: The statement is true for all natural numbers. 4