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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

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4 Sets and Mappings

Sets and Mappings

4.1

Elements of Set Theory

A set is any collection of elements. Sets can be defined . . . • . . . by enumeration of their elements, e.g., S = {2, 4, 6, 8}, or • . . . by description of their elements, e.g., S = {x|x is a positive even integer greater than zero and less than 10}.

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

When we wish to denote membership or inclusion in a set, we use the symbol ∈. E.g., if S = {2, 5, 7}, we say that 5 ∈ S . A set S is a subset of another set T if every element of S is also an element of T . • We write S ⊂ T (S is contained in T ) or • T ⊃ S (T contains S). If S ⊂ T , then x ∈ S ⇒ x ∈ T .

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Two sets are equal sets if they each contain exactly the same elements. • We write S = T whenever x ∈ S ⇒ x ∈ T and x ∈ T ⇒ x ∈ S . • Thus, S and T are equal sets if and only if S ⊂ T and T ⊂ S . E.g. , if S = {integers, x|x2 = 1} and T = {−1, 1}, then S = T . A set S is empty or is an empty set if it contains no elements at all. • E.g., if A = {x|x2 = 0 and x > 1}, then A is empty. • We denote the empty set by the symbol ∅ and write A = ∅.

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

The complement of a set S in a universal set U is the set of all elements in U that are not in S and is denoted S c . • If U = {2, 4, 6, 8} and S = {4, 6}, then S c = {2, 8}. • More generally, for any two sets S and T in a universal set U , we define the set difference denoted S \ T , or S − T , as all elements in the set S that are not elements of T . • Thus, we can think of the complement of the set S in U as the set difference S c = U − S .

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

The basic operations on sets are union and intersection. They correspond to the logical notions of “or” and “and,” respectively. • For two sets S and T , we define the union of S and T as the set S ∪ T ≡ {x|x ∈ S or x ∈ T }. • We define the intersection of S and T as the set S ∩ T ≡ {x|x ∈ S and x ∈ T }.

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

Figure 4-1: Different Sets (JR-A1.1)

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4 Sets and Mappings

Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Sets constructed from other sets • collect the necessary (possibly infinite) number of integers starting with 1 into an index set, I ≡ {1, 2, 3, ...} • denote the collection of sets more simply as {Si }i∈I . • denote the union of all sets in the collection by ∪i∈I Si • denote the intersection of all sets in the collection as ∩i∈I Si .

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

The product of two sets • product of two sets S and T is the set of “ordered pairs” in the form (s, t) • the first element in the pair is a member of S and the second is a member of T • the product of S and T is denoted S × T ≡ {(s, t)|s ∈ S, t ∈ T }

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Cartesian plane • familiar set product: “Cartesian plane.” • the plane in which we commonly graph things • visual representation of a set product constructed from the set of real numbers • the set of real numbers is denoted by the special symbol R and is defined as R ≡ {x| − ∞ < x < ∞} • consider the set product R × R ≡ {(x1 , x2 )|x1 ∈ R, x2 ∈ R} where any point in the set (any pair of numbers) can be identified with a point in the Cartesian plane depicted • R × R is sometimes called “two-dimensional Euclidean space” and is commonly denoted R2 . 4-9

Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

Figure 4-2: Cartesian Plane (JR-A 1.2.)

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4 Sets and Mappings

Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Euclidean space, or “n-space” • any n-tuple, or vector, is just an n-dimensional ordered tuple (x1 , ..., xn ) • a “point” in n-dimensional Euclidean space, or “n-space.” • n-space is defined as the set product Rn ≡ R × R×, · · · × R ≡ {(x1 , ..., xn )|xi ∈ R, i = 1, ..., n}. | {z } n times • usually denote vectors or points in Rn with boldface type, so that x ≡ (x1 , ..., xn ) • Often, we restrict attention to a subset of Rn , called the n with “nonnegative orthant,” denoted R+ Rn+ ≡ {(x1 , ..., xn )|xi ≥ 0, i = 1, ..., n} ⊂ Rn 4-11

Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Comparing Vectors n • x ≥ 0 indicates vectors in R+ , where each component xi is greater than or equal to zero

• x >> 0 indicates vectors where every component of the vector is strictly positive. • More generally, for any two vectors x and y in Rn , we say that x ≥ y iff xi ≥ yi , i = 1, ..., n. We say that x >> y if xi > yi , i = 1, ..., n.

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Definition 4.1 (Convex Sets in Rn ). S ⊂ Rn is a convex set if for all x1 ∈ S and x2 ∈ S, we have tx1 + (1 − t)x2 ∈ S

∀t ∈ [0, 1].

• i.e., a set is convex if for any two points in the set, all weighted averages of those two points (where the weights sum to 1) are also points in the same set • the weighted average used in the definition is called a convex combination – z is a convex combination of x1 and x2 if z = tx1 + (1 − t)x2 for some number t ∈ [0, 1]; z is thus a point that, in some sense, “lies between” the two points x1 and x2 .

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

• Consider x1 ∈ R and x2 ∈ R, where x1 = 8 and x2 = 2.

• The convex combination, z = tx1 + (1 − t)x2 can be multiplied out and rewritten as z = x2 + t(x1 − x2 ). • Take x2 as the “starting point,” and the difference (x1 − x2 ) as the “distance” from x2 to x1 : – This expression says that z is a point located at the spot x2 plus some proportion t, less than or equal to 1, of the distance between x2 and x1 .

• As long as we choose a value of t in the interval 0 ≤ t ≤ 1, the convex combination will always lie somewhere strictly in between the two points, or it will coincide with one of the points. 4-14

Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Figure 4-3: Convex combination - example (JR-A.1.3)

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Notice • in the above example, notice that x1 , x2 , and every point in between x1 and x2 can be expressed as the convex combination of x1 and x2 for some value of t between zero and 1 • the set of all convex combinations of x1 and x2 will therefore be the entire line segment between x1 and x2 , including those two points • these basic ideas carry over to sets of points in two dimensions as well

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

look again at the definition of a convex set: Definition 4.2 (Convex Sets in Rn ). S ⊂ Rn is a convex set if for all x1 ∈ S and x2 ∈ S, we have tx1 + (1 − t)x2 ∈ S

∀t ∈ [0, 1].

Note: We could as well have said that a set is convex if it contains all convex combinations of every pair of points in the set. ⇒ simple and intuitive rule defining convex sets:

Definition 4.3. A set is convex iff we can connect any two points in the set by a straight line that lies entirely within the set. Convex sets are all “nicely behaved.” They have no holes, no breaks, and no awkward curvatures on their boundaries. 4-17

Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Figure 4-4: Examples of convex and non-convex sets (JR-A.1.5)

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Let S and T be convex sets in Rn . Then S ∩ T is a convex set. Proof: Let S and T be convex sets. Let x1 and x2 be any two points in S ∩ T .

Because x1 ∈ S ∩ T, x1 ∈ S and x1 ∈ T . Because x2 ∈ S ∩ T, x2 ∈ S and x2 ∈ T .

Let z = tx1 + (1 − t)x2 for t ∈ [0, 1] be any convex combination of x1 and x2 . Because S is a convex set, z ∈ S. Because T is a convex set, z ∈ T. Because z ∈ S and z ∈ T , z ∈ S ∩ T . Because every convex combination of any two points in S ∩ T is also in S ∩ T , S ∩ T is a convex set. QED

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4.2

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Relations and Functions

4 Sets and Mappings

Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Ordered pairs • We have seen that any ordered pair (s, t) associates an element s ∈ S to an element t ∈ T . • The sets S and T need not contain numbers; they can contain anything. • Any collection of ordered pairs is said to constitute a binary relation between the sets S and T .

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

• – E.g., let S be the set of cities {Washington, London, Marseilles, Paris}, and T be the set of countries {United States, England, France, Germany}.

– The statement “is the capital of ” then defines a relation between these two sets that contains the elements {(Washington, United States), (London, England), (Paris, France)}. • • When s ∈ S bears the specified relationship to t ∈ T , we denote membership in the relation R in one of two ways: – (s, t) ∈ R – sRt.

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Relations on a set • Many familiar binary relations are contained in the product of one set with itself. – E.g., let S be the closed unit interval, S = [0, 1]. – Then the binary relation ≥, illustrated in Fig. 4-5, consists of all ordered pairs of numbers in S where the first one in the pair is greater than or equal to the second one. • A binary relation is a subset of the product of one set S with itself, it is a relation on the set S .

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Figure 4-5: Example - binary relation ≥ (JR-A.1.6)

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Definition 4.4 (Completeness). A relation R on S is complete if, for all elements x and y in S, xRy or yRx. • Suppose that S = {1, 2, ..., 10}, and consider the relation defined by the statement, “is greater than.” • This relation is not complete because one can easily find some x ∈ S and some y ∈ S where it is neither true that x > y nor that y > x: – E.g. , one could pick x = y = 1, or x = y = 2, and so on.

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

• The definition of completeness does not require the elements x and y to be distinct, so nothing prevents us from choosing them to be the same. • Because no integer can be either less than or greater than itself, the relation “is greater than” is not complete. • However, the relation on S defined by the statement “is at least as great as” is complete. – For any two integers, whether distinct or not, one will always be at least as great as the other, as required by completeness.

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Definition 4.5 (Transitivity). A relation R on S is transitive if, for any three elements x, y, and z in S, xRy and yRz implies xRz . • Both relations considered above are transitive.

– If x is greater than y and y is greater than z, then x is certainly greater than z .

– The same is true for the relation defined by the statement “is at least as great as.” • But the relation “is a friend of ” is not transitive.

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Functions • A function is a common, though very special kind of relation. • Specifically, a function is a relation that associates each element of one set with a single, unique element of another set. • We say that the function f is a mapping from one set D to another set R and write f : D → R. •

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• If y is the point in the range mapped into by the point x in the domain, we write y = f (x). • To denote the entire set of points A in the range that is mapped into by a set of points B in the domain, we write A = f (B). 4-28

Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Figure 4-6: Example - not a function (JR-A.1.7(a))

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Figure 4-7: Example - a function (JR-A.1.7(b))

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

• The of f is that set of points in the range into which some point in the domain is mapped, i.e., I ≡ {y|y = f (x), for some x ∈ D} ⊂ R. • The of a set of points S ⊂ I is defined as f −1 (S) ≡ {x|x ∈ D, f (x) ∈ S}. • The f is the set of ordered pairs G ≡ {(x, y )|x ∈ D, y = f (x)}.

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

• There is nothing in the definition of a function that prohibits more than one element in the domain from being mapped into the same element in the range (see Fig. 4-7). • If every point in the range is assigned to at most a single point in the domain, the function is called one-to-one. • If the image is equal to the range – if every point in the range is mapped into by some point in the domain – the function is said to be onto. • If a function is one-to-one and onto, then an inverse function f −1 : R → D exists that is also one-to-one and onto.

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4.3

4 Sets and Mappings

Some Topology

Definition 4.6 (Topology). Topology is the study of fundamental properties of sets and mappings. We introduce a few basic topological ideas and use them to establish some important results about sets, and about continuous functions from one set to another. Most of the ideas discussed here may be generalized to arbitrary types of sets, we confine ourselves to considering sets in Rn , i.e., sets that contain real numbers or vectors of real numbers.

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

• A metric is a measure of distance. • A metric space is a set with a notion of distance defined among the points within the set. – E.g., the real line, R, together with an appropriate function measuring distance, is a metric space. • One such distance function, or metric, is just the absolute-value function. For any two points x1 and x2 in R, the distance between them, denoted d(x1 , x2 ), is given by d(x1 , x2 ) = |x1 − x2 | • The Cartesian plane, R2 , is also a metric space. 4-34

Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

A natural notion of distance defined on the plane is inherited from Pythagoras. • Choose any two points x1 = (x11, x21 ) and x2 = (x21 , x22 ) in R2 . • Construct the right triangle connecting the two points where the horizontal leg is of length a and the vertical leg is length b. • Pythagoras tells us the length of the hypotenuse is equal to √ a2 + b2 or q p 1 2 2 2 d(x , x ) = a + b = (x12 − x11 )2 + (x22 − x21 )2 .

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

Both these distance formulas are special cases of the same formula: • In R, the absolute value |x1 − x2 | can be expressed as p (x1 − x2 ) · (x1 − x2 ). • Hence, we could write p 1 2 1 2 d(x , x ) = |x − x | = (x1 − x2 )(x1 − x2 ).

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

• In R2

– we can first apply the rules of vector subtraction to obtain (x1 − x2 ) = (x11 − x12 , x21 − x22 )T ,

– then apply the rules of vector (dot) multiplication to multiply this difference with itself: (x 1 − x 2 ) · (x 1 − x 2 ) = = = =

(x11 − x12 , x21 − x22 ) · (x11 − x21 , x21 − x22 )T

(x11 − x12 )2 + (x21 − x22 )2

(−1)2 (x12 − x11 )2 + (−1)2 (x22 − x21 )2

(x21 − x11 )2 + (x22 − x12 )2 .

– Pythagoras tells us that we can measure the distance between two points as the square root of a product of the difference between the two points, this time the vector product of the vector difference of two points in the plane. 4-37

Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

– Analogous to above, we can therefore write p d(x1 , x2 ) = (x1 − x2 ) · (x1 − x2 ) for x1 and x2 in R2 . • The distance between any two points in Rn is just a direct extension of these ideas. • In general, for x1 and x2 in Rn , p d(x1 , x2 ) ≡ (x 1 − x 2 ) · (x 1 − x 2 ) q ≡ (x11 − x21 )2 + (x21 − x22 )2 + · · · + (xn1 − xn2 )2 , • We use the notation d(x1 , x2 ) ≡ kx1 − x2 k. • This formula is the Euclidean metric or Euclidean norm. • Metric spaces Rn that use this as the measure of distance are called Euclidean spaces.

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

• Now we can make precise what it means for points to be “near” each other. – If we take any point x0 ∈ Rn , we define the set of points that are less than a distance ǫ > 0 from x0 as the open ǫ-ball with center x0 . – The set of points that are a distance of ǫ or less from x0 is called the closed ǫ-ball with center x0 .

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Mathematics for Economists (Master Applied, Winter 2018/19) Dr. Johannes Maier

4 Sets and Mappings

. 1. The open ǫ-ball with center x0 and radius ǫ > 0 (a real number) is the subset of points in Rn : Bǫ (x0 ) ≡ {x ∈ Rn |

d(x0 , x) < ǫ } | ...