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Brief Review of Some Basic Logic and Calculus

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Brief Review of Some Basic Logic and Calculus Mathematical methods for economic theory: a tutorial by Martin J. Osborne Copyright © 1997-2007 Martin J. Osborne. Version: 2007/7/23. 1.1 Logic Basics When making precise arguments, we often need to make conditional statements, like if the price of output increases then a competitive firm increases its output or if the demand for a good is a decreasing function of the price of the good and the supply of the good is an increasing function of the price then an increase in supply at every price decreases the equilibrium price. These statements are instances of the statement if A then B, where A and B stand for statements. We may alternatively write this general statement as A implies B, or, using a symbol, as A ⇒ B. Yet two more ways in which we may write the same statement are A is a sufficient condition for B, and B is a necessary condition for A. (Note that B comes first in the second of these two statements!!) Important note: The statement A ⇒ B does not make any claim about whether B is true if A is NOT true! It says only that if A is true, then B is true. While this point may seem obvious, it is sometimes a source of error, partly because we do not always adhere to the rules of logic in everyday communication. For example, when we say " if it's fine tomorrow then let's play tennis" we probably mean both " if it's fine tomorrow then let's play tennis" and " if it's not fine tomorrow then let's not play tennis" (and maybe also " if it's not clear whether the weather is good enough to play tennis tomorrow then I'll call you"). When we say "if you listen to the radio at 8 o'clock then you'll know the weather forecast", on the other hand, we do not mean also " if you don't listen to the radio at 8 o'clock then you won't know the weather forecast", because you might listen to the radio at 9 o'clock or check on the web, for example. The point is that the rules we use to attach meaning to statements in everyday language are subtle, while the rules we use in logical arguments are absolutely clear: when we make the logical statement " if A then B", that's exactly what we mean---no more, no less. We may also use the symbol " ⇐" to mean "only if" or "is implied by". Thus B ⇐ A is equivalent to A ⇒ B. Finally, the symbol " ⇔" means "implies and is implied by", or "if and only if". Thus A ⇔ B is equivalent to A ⇒ B and A ⇐ B. If A is a statement, we write the claim that A is not true as not(A). If A and B are statements, and both are true, we write A and B, and if at least one of them is true we write A or B. Note, in particular, that writing "A or B" includes the possibility that both statements are true. Two rules Rule 1 If the statement A ⇒ B is true, then so too is the statement (not B) ⇒ (not A). The first statement says that whenever A is true, B is true. Thus if B is false, A must be false---hence the second statement. Rule 2 The statement not(A and B) is equivalent to the statement (not A) or (not B). Note the "or" in the second statement! If it is not the case that both A is true and B is true (the first statement), then either A is not true or B is not true. Quantifiers We sometimes wish to make a statement that is true for all values of a variable. For example, denoting the total demand for tomatoes at the price p by D(p), it might be true that D(p) > 100 for every price p in the set S. In this statement, "for every price" is a quantifier . Important note : We may use any symbol for the price in this statement: " p" is a dummy variable. After having defined D(p) to be the total demand for tomatoes at the price p, for example, we could write D(z) > 100 for every price z in the set S. Given that we just used the notation p for a price, switching to z in this statement is a little odd, BUT there is absolutely nothing wrong with doing so! In this simple example, there is no reason to switch notation, but in some more complicated cases a switch is unavoidable (because of a clash with other notation) or convenient. The point is that every statement of the form A( x) for every x in the set Y, where x is any symbol, has exactly the same content. Another type of statement we sometimes need to make is A( x) for some x in the set Y, or, equivalently, there exists x in the set Y such that A( x). "For some x" (alternatively "there exists x") is another quantifier , like "for every x"; my comments about notation apply to it. Exercises Copyright © 2000-2003 by Martin J. Osborne 1.2 Matrices and solutions of systems of simultaneous equations Matrices I assume that you are familiar with vectors and matrices and know, in particular, how to multiply them together. (Do the first few exercises to check your knowledge.) The determinant of the 2 × 2 matrix a b c d is ad − bc. If ad − bc ≠ 0, the matrix is nonsingular ; in this case its inverse is 1 d −b . ad − bc −c a (You can check that the product of the matrix and its inverse is the identity matrix.) The determinant of the 3 × 3 matrix a b c d e f g h i is ∆ = a(ei − h f ) − b(di − g f ) + c(dh − eg). If ∆ ≠ 0 the matrix is nonsingular ; in this case its inverse is D −D D 1 11 12 13 −D21 D22 −D23 ∆ D31 −D32 D33 where Dij is the determinant of the 2 × 2 matrix obtained by deleting the ith column and jth row of the original 3 × 3 matrix. That is, D11 = ei − h f , D12 = bi − ch, D13 = b f − ec, D21 = di − f g, D22 = ai − cg, D23 = a f − dc, D31 = dh − eg, D32 = ah − bg, and D33 = ae − db. (Again, you can check that the product of the matrix and its inverse is the identity matrix.) The determinant of the matrix A is denoted A. Let A be an n × m matrix (i.e. a matrix with n rows and m columns). The transpose A′ of A is the m × n matrix in which, for i = 1, ..., n, the ith row of A′ is the ith column of A. In particular, if x is a column vector (1 × n matrix) then x′ is a row vector ( n × 1 matrix). Solutions of systems of simultaneous equations Consider a system of two equations in two variables x and y: ax + by = u cx + dy = v. Here are three ways to solve for x and y. 1. Isolate one of the variables in one of the equations and substitute the result into the other equation. For example, from the second equation we have y = ( v − cx)/ d. Substituting this expression for y into the first equation yields ax + b(v − cx)/ d = u, which we can write as (a − bc/d)x + bv/d = u, so that u − bv/d x = a − bc/d or ud − bv x = . ad − bc To find y we now use the fact that y = ( v − cx)/ d, to get va − cu y = . ad − bc 2. Use Cramer's rule (due to Gabriel Cramer, 1704-1752 ). Write the two equations in matrix form as a b x u = (*) c d y v Cramer's rule says that the solutions are given by u b x = v d ad − bc and a u y = c v . ad − bc Note that the denominator in these expressions, ad − bc, is the determinant of the matrix at the left of (*). The matrix in the numerator of each expression is obtained by replacing the column in the matrix on the left of (*) that corresponds to the variable for which we are solving with the column vector on the right of (*). Calculating the determinants in the numerators, we have ud − bv x = ad − bc and va − cu y = . ad − bc 3. Write the two equations in matrix form as a b x u = c d y v (as when using Cramer's rule) and solve by inverting the matrix on the left hand side. The inverse of this matrix is 1 d −b ad − bc −c a so we have x 1 d −b u = y ad − bc −c a v so that ud − bv x = ad − bc and va − cu y = . ad − bc Which of these three methods is best depends on several factors, including your ability to remember Cramer's rule and/or how to invert a matrix. If you have to solve for only one of the variables, Cramer's rule is particularly convenient (if you can remember it). For a system of more than two variables, the same three methods are available. The first method, however, is pretty messy, and unless you are adept at inverting matrices, Cramer's rule is probably your best bet. Exercises Copyright © 1997-2003 by Martin J. Osborne 1.3 Intervals and functions Intervals An interval is a set of (real) numbers between, and possibly including, two numbers.
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