Sets and functions 1 Sets The language of sets and functions pervades mathematics, and most of the important operations in mathematics turn out to be functions or to be ex- pressible in terms of functions. We will not define what a set is, but take as a basic (undefined) term the idea of a set X and of membership x 2 X (x is an element of X). The negation of x 2 X is x2 = X: x is not an element of X. Typically, the elements of a set will themselves be sets, underscoring the point that, in mathematics, everything is a set. A set can be described (i) as a list fx1; : : : ; xng or (ii) by giving a description of its elements, e.g. the set of positive real numbers is described via fx 2 R : x > 0g; where R denotes the set of all real numbers. A very important set is the empty set ;: for all x, x2 = X. Thus ; has no elements. Two sets X and Y are by definition equal if they have the same elements: X = Y if, for all x, x 2 X () x 2 Y . We can say this somewhat informally as follows: a set is uniquely specified by its elements. This implies by logic that ; is uniquely specified by the condition that, for all x, x2 = X: there is exactly one empty set. If, for every x 2 X; x 2 Y , then X is a subset of Y , written X ⊆ Y . Note that we always have X ⊆ X and ; ⊆ X. A subset A of X is called a proper subset if A 6= X. By the definition of equality of sets, X = Y () X ⊆ Y and Y ⊆ X. If X ⊆ Y and Y ⊆ Z, then X ⊆ Z; this is called the transitivity property. The notation X ⊂ Y is sometimes used to mean that X ⊆ Y but X 6= Y . A set of the form X = fx1; : : : ; xng is a finite set. If for all i; j with 1 ≤ i; j ≤ n, we have xi 6= xj then we write #(X) = n. By logic or convention, ; is finite and #(;) = 0. Conversely, if X is a (finite) set with #(X) = 0, then X = ;. A set of the form fxg has exactly one element. In particular, f;g has a single element, namely ;, and thus f;g 6= ;. Recall the standard operations on sets: 1 Definition 1.1. If X1 and X2 are two sets, then: 1. The union of X1 and X2 is the set X1 [ X2 = fx : x 2 X1 or x 2 X2g: Thus X1 ⊆ (X1 [X2) and X2 ⊆ (X1 [X2). The union of finitely many sets is defined similarly: If X1;:::;Xn are sets, then n [ Xi = fx : for some i, x 2 Xig: i=1 2. The intersection of X1 and X2 is: X1 \ X2 = fx : x 2 X1 and x 2 X2g: Thus (X1 \ X2) ⊆ X1 and (X1 \ X2) ⊆ X2. Likewise n \ Xi = fx : for all i, x 2 Xig: i=1 3. Given two sets X1 and X2, the complement of X2 in X1, written X1 − X2, is the set fx 2 X1 : x2 = X2g: Thus X2 \ (X1 − X2) = ;. If X2 ⊆ X1, then X2 [ (X1 − X2) = X1. For example, X − X = ; and X − ; = X. By logic (deMorgan's laws), Y − (X1 \ X2) = (Y − X1) [ (Y − X2) and Y − (X1 [ X2) = (Y − X1) \ (Y − X2). (For example, if x 2 Y; x2 = X1 \ X2, then either x2 = X1 or x2 = X2 and conversely.) Definition 1.2. Given X and Y , we define X ×Y , the Cartesian product of X and Y , to be the set of ordered pairs (x; y) with x 2 X and y 2 Y . Here x is the first component or first coordinate of the ordered pair (x; y) and y is the second component (or coordinate). Clearly, if A ⊆ X and B ⊆ Y , then A × B ⊆ X × Y . If X = Y , we abbreviate X × X by X2. Likewise, if we have n sets X1;:::;Xn, then X1 × · · · × Xn is the set of ordered n-tuples (x1; : : : ; xn) th with xi 2 Xi for every i, the i component (or coordinate) of (x1; : : : ; xn) n is xi, and we again abbreviate X × · · · × X (n times) by X . 2 Remark 1.3. The operative properties of an ordered pair (x; y) are: 1) For all x 2 X and y 2 Y , there exists an ordered pair (x; y) 2 X × Y , and 2) two ordered pairs (x1; y2) and (x2; y2) are equal () they have the same first components and the same second components, i.e. () x1 = x2 and y1 = y2; it is not enough to require that the sets fx1; y1g and fx2; y2g be equal. It is possible to give a formal definition of an ordered pair just using set theory. In fact, one can define (x; y) = ffxg; fx; ygg. (In other words, ordered pair does not have to be an undefined term.) However, we shall not really care what the precise definition is, but only that an ordered pair has the operative properties 1) and 2) above. Using functions, though, we can give a careful definition of an ordered n-tuple; we shall describe this later. If X and Y are finite sets, then X × Y is also finite, and #(X × Y ) = #(X)#(Y ); and similarly for the product of n finite sets X1 × · · · × Xn. In particular, this formula says that #(; × X) = #(X × ;) = 0 for every (finite) set X and hence that, if X is finite, then ; × X = X × ; = ;. Of course, it is easy to check by logic that ; × X = X × ; = ; for every set X (not necessarily finite). The set of all subsets of X is also a set, and is called the power set of X, often denoted P(X): P(X) = fA : A ⊆ Xg: By the transitivity property, if Y ⊆ X, then P(Y ) is a subset of P(X). Note that X 2 P(X) and that ; 2 P(X). If X 6= ; and x 2 X, then fxg 2 P(X). Examples: P(;) = f;g. In particular, P(;) 6= ;; in fact, P(;) contains the unique element ;, and thus #(P(;)) = 1. Likewise, P(P(;)) = P(f;g) = f;; f;gg. In particular, #(P(P(;))) = 2. Likewise, P(f;; f;gg) = f;; f;g; ff;gg; f;; f;ggg; and hence #(P(f;; f;gg)) = 4: More generally, we shall see that, if X is a finite set and #(X) = n, then #(P(X)) = 2n. 2 Functions Next we define a function f : X ! Y . Although we can think of a function as a \rule" which assigns to every x 2 X a unique y 2 Y , it is easier to make 3 this concept precise by identifying the function f with its graph in X × Y (as we were taught not to do in calculus). Thus a function f is the same thing as a subset Gf of X × Y with the following property: for all x 2 X, there is a unique y 2 Y such that (x; y) 2 Gf and we set y = f(x). To say that there is a unique y 2 Y says that f(x) is uniquely determined by x, and to say that for every x 2 X there exists an (x; y) 2 Gf says that in fact f(x) is defined for all x 2 X. This is the so-called vertical line test: for each x 2 X, we have the subset fxg × Y of X × Y . (In case X = Y = R, such subsets are exactly the vertical lines.) Then G is the graph of a function f if and only if, for every x 2 X,(fxg × Y ) \ G consists of exactly one point, necessarily of the form (x; y) for some y 2 Y . The unique such y is then f(x). In the above notation, we call X the domain of f and Y the range. Thus the domain and range are a part of the information of a function. Note that a function must be defined at all elements of its domain; thus for example the function f(x) = 1=x cannot have domain R without assigning some value to f(0). (This is in contrast to the practice in some calculus courses where f is allowed to be not everywhere defined.) Two functions f1 and f2 are equal if and only if their graphs are equal, if and only if, for all x 2 X, f1(x) = f2(x). Thus, just as a set is specified by its elements, a function is uniquely specified by its values. We emphasize, though, that for two functions f1 and f2 to be equal, they must have the same domain and range. We shall use the word map or mapping as a synonym for function; typically maps are functions in some kind of geometric setting. Definition 2.1. Let f : X ! Y be a function. The set fy 2 Y : there exists x 2 X such that f(x) = yg is called the image of f and is sometimes written f(X). (Sometimes people call the range the codomain and define the range to be the image.) More generally, if A is a subset of X, then we set f(A) = fy 2 Y : there exists an x 2 A such that f(x) = yg: We say that f is surjective or onto if f(X) = Y , in other words if the image of f is Y . In general the image of a function f(x) will be a subset of the range, but need not equal the range.