A bit of notation

Mathematics and logic have quite a bit of notation. We shall stick to what saves enough time to be worthwhile. You are free to use more on an exam (like the following show up quite often: \(\therefore\) resp. \(\because\) for "therefore" resp. "because" and \(\wedge\) resp. \(\vee\) for "and simultaneously" resp. "or"), but graders value content over form. 

Logic: 

• \(\Rightarrow\) means implication: if (LHS) then (RHS). \(f(x)>1\Rightarrow f(x)\geq 0\) is a true statement. 
  \(\Leftarrow\) means implication the other way around. 
  (Logicians distinguish between, e.g. "entails", "claims implication", "proves implication" etc. - don't worry.) 
• Note about language: We shall encounter conditions like \(\quad\lambda\geq 0\quad(=0 \text{ if }g<b)\quad\)
  That means that lambda is nonnegative and furthermore that if g<b then lambda must be zero (not the other way around! And g=b does not rule out lambda being zero.)
• \(\Leftrightarrow\) means logical equivalence, that is often abbreviated "iff" for "if and only if". 
  That means that both the left-hand statement implies the right-hand AND the other way around.
  Example: \(x\cdot(x-1)=0\quad\Leftrightarrow\quad x=0\text{ or }x=1.\)
  (Note "or", because logical "and" means "and simultaneously", and we cannot have x=0 simultaneously as x=1. Nevertheless, I (Nils) as grader won't penalize an "and" there if it is clear what you mean (and you mean the right thing!) - because we do say that x(x-1)=0 has solutions x=0 and x=1. I won't answer for other graders, though.)

Sets and such:

• \(x\in S\): x (an element: point, number, ...) is member of the set S.
  Sometimes also: \(S\ni x\) for S owns x.
  Strikeout to say "not a member of": \(x\not\in U\)
• "on" a set: throughout the entire set. Saying \(f(x)=0\) in S would mean that f has a zero in S; saying \(f(x)=0\) on S, means it is zero for every \(x\in S\). 
• The empty set is commonly denoted \(\varnothing\). You might see \(\neq\varnothing\) to indicate that a set is not empty.
• "open" set: contains none of its boundary points (if it has any). \(\varnothing\) is open. 
  "closed" set: contains all of its boundary points (if it has any). \(\varnothing\) is closed as well as open. The real line is closed too. 
• Intervals: Let b>a. Then we use brackets to indicate that the endpoint is included, and parenthesis to indicate it is not: \([a,b]\) is the closed interval from & including a, to & including b. \([a, b)\) means that a is included and b is not. We use \((-\infty,b]\) for the (closed!) interval up to & including b, and \((a,+\infty)\) for the (open!) interval from-but-not-including a and up. 
• \(\mathbf N\) (book, in handwriting we often use \(\mathbb N\)): the set of natural numbers 1, 2, ... . Note, infinity is not a number. 
  Convention: 0 is not included.
  _{(In computer science - and France - they do include 0. Don't do it in this course, if it causes confusion.)}
• \(\mathbf R\) (book; \(\mathbb R\) in handwriting): The real line. I.e. the set of real numbers. Again, infinity is not a number. 
  _{(Aside: Why the word "real"? Lots of maths require the "imaginary" unit \(i \)that satisfies \(i^2=-1\) - that is "imaginary", as opposed to "real")}
• \(\mathbf R^n \) (book; \(\mathbb R^n\) in handwriting): The space of n-tuples \((x_1,\dots,x_n)\), each \(x_i \) being a real number. 
• "Vector notation" in functions:  \(\mathbf x\) (I'll use \(\vec x\) on the board) for \((x_1,\dots,x_n)\) (more when we come to linear algebra) as shorthand notation for arguments of a function of n variables. 
• \(A\subseteq B\): the set B contains every element of the set A (and possibly more).
  Sometimes also: \(B\supseteq A\)
  Example use: Speaking of a function f on a set \(S\subseteq \mathbf R^n\), implicitly means that f is a function of n variables. 
• "Set builder" notation: \(S=\{\mathbf x\in \mathbf R^n; \ p_1 x_1+\dots+p_n x_n=m\}\) for the set of n-tuples \(\mathbf x=(x_1,\dots,x_n)\) which satisfies everything after the semicolon, in this case the budget constraint. 
  "More common": vertical bar in place of semicolon, but often worse to read when handwritten.
• A couple of pieces of notation hardly used in Math 2 (but seen in the micro course):
  • The Cartesian product of two sets \(\mathbf U\times\mathbf V\): pairs \((u,v)\) (or, frequently, \((\mathbf u,\mathbf v)\) such that \(u\in U\) and \(v\in V\).
    For example, my payoff in a game is a function of my action u and your action v, where u is taken from my action space U and v from yours V.
    Note: \(\mathbf R^n\) is the Cartesian product \(\mathbf R\times\mathbf R\times\dots\mathbf R\)
  • \(\mathbf R_+\) for nonnegatives, \(\mathbf R^n_+\) when all the \(x_i\) are \(\geq 0\), \(\mathbf R^n_-\) when all the \(x_i\) are \(\leq 0\). Example: production possibility sets. m inputs (by convention negative, as they are spent in production) and n outputs (positive). A possibility set is then some \(S\subseteq\mathbf R^m_-\times \mathbf R^n_+\)and \((\mathbf x,\mathbf y)\in S\) if \(\mathbf y\) can be produced from \(\mathbf x\)

Functions and such

• A function \(f\) of n variables is \(C^k\) if f itself and all its partial derivatives of order 1, ..., k, are continuous functions. 
  (This is actually a "set of functions", but I list it here under functions.)
• Partial derivatives: For \(F(K,L)\), this course uses \(\frac{\partial F}{\partial K}\) or \(F'_K\) or \(F'_1\) for the partial derivative wrt. the first variable. We avoid the notation from partial differential equations to use merely subscript without prime. 
  You might sometimes see \(\nabla F\) for the n-tuple \((F'_1,\dots,F'_n)\), but I will try to avoid it.
• Elasticities issue: If you are on an isoquant C and that defines, L as function of K, then \(\frac{d}{dK}F(K,L(K))\) means the total derivative, namely \(\frac{\partial F}{\partial K}+\frac{\partial F}{\partial L}\frac{dL}{d K}\) (which equals zero and gives rise to the MRS formula!). For derivatives we have two symbols \(d\) for ordinary/total differentiation and \(\partial \) for partial. For elasticities, does \(\text{El}_KF(K,L(K))\) mean partial or total? (Most often partial!)

Misc ... limits ... 

• When we write \(\displaystyle \lim_{x\to a}\) we adopt the notational convention that x does not hit a. Otherwise we would have had to write \(\displaystyle \lim_{h\to 0,\ h\neq 0}\frac{f(x+h)-f(x)}h\)
• \(\displaystyle \lim_{x\to a^+}\) for x goes to a from the right (from above), \(\displaystyle \lim_{x\to a^-}\) for limit from the left. Sometimes omitted when it isn't well-defined the other way (you will see \(\displaystyle \lim_{x\to 0}\ln x=-\infty\) in the literature).
• Did you see the latter "\(=-\infty\)"? We write that for limits even if infinity is not a number. A bit inconsistent: we say that \(\displaystyle \lim_{x\to 0^+}\: \ln x\) does not exist, yet we write it as \(=-\infty. \) It diverges to \(-\infty\)  , not converges.
• Also, you will see e.g. \(f(0^+)\) for \(\displaystyle \lim_{x\to 0^+}f(x)\)
• Although I shall try to specify the "+" in \(\displaystyle \lim_{x\to+\infty}\)when x goes to positive infinity, it is fairly common in the literature to omit it if it is "clear" from the context - and even more, that "\(n\to\infty\)" means that n runs through sequences of natural numbers.