Review of Functions Functions Philippe B. Laval KSU Current Semester Philippe B. Laval (KSU) Functions Current Semester 1 / 12
Introduction Students are expected to know the following concepts about functions: s and notation. Operations on functions One-to-one and onto functions Inverse of a function You may review these concepts in the notes provided. We will only discuss in details the direct image and inverse image of a set under a function. Philippe B. Laval (KSU) Functions Current Semester 2 / 12
Review: s Let A and B denote two sets. 1 A function f from A to B is a rule which assigns a unique y B to each x A. We write y = f (x). f (x) denotes the value of the function at x. 2 x is called the independent variable (also called an input value), y is the dependent variable (also called an output value). Remember, if we say that y is a function of x, it implies that it depends on x. 3 The domain of f is A, the set of values of x. It is also denoted D (f ) or Dom f. When the domain of a function is not given, it is understood to be the largest set of real numbers for which the function is defined. 4 The range of f is the set R (f ) = {f (x) : x D (f )}. It is also denoted Range f. Philippe B. Laval (KSU) Functions Current Semester 3 / 12
Review: s Let A and B denote two sets. A function from A to B is a subset of A B that is a set of ordered pairs with the property that whenever (a, b) f and (a, c) f then b = c. If f is a function from A to B and y = f (x) (or if (x, y) f ), then we say that y is the image of x under f. If f is a function from A to B, we also say that f is a mapping from A into B, or that f maps A into B. We often write: f : A B Philippe B. Laval (KSU) Functions Current Semester 4 / 12
Review: Onto, and one-to-one functions If f is a mapping of A into B such that R (f ) = B, then we say that the mapping is onto. We also say that f is surjective, or that f is a surjection. Let f be a function from A into B. f is said to be injective, or an injection, or one-to-one if one of the three equivalent conditions below is satisfied. 1 f (a) = f (b) a = b 2 a b f (a) f (b) 3 (a, c) f and (b, c) f a = b Philippe B. Laval (KSU) Functions Current Semester 5 / 12
Review: Bijection A function which is both an injection (one-to-one) and a surjection (onto) is called a bijection. Philippe B. Laval (KSU) Functions Current Semester 6 / 12
Review: Inverse Function Let f be an injective function from A onto B. The function f 1 = {(b, a) B A : (a, b) f } is called the inverse of f. It is denoted f 1. f and f 1 are related in many different ways. We list a few of these relations below. 1 D (f ) = R ( f 1), R (f ) = D ( f 1). This can be seen from the definition. 2 y = f (x) x = f 1 (y). This can be seen from the definition. ( 3 f 1 f ) (x) = x x D (f ). See problems at the end of this section. ( 4 f f 1 ) (y) = y y R (f ). See problems at the end of this section. Philippe B. Laval (KSU) Functions Current Semester 7 / 12
Direct Image of a Set: s and Examples Let f : A B and let E be a set such that E A. The direct image of E, denoted f (E), is defined by: Example f (E) = {f (x) : x E} Remark It should be clear to the reader that f (E) B. Consider f : N N defined by f (n) = n 2 1. Let E = {2, 3, 4}. Find f (E). Philippe B. Laval (KSU) Functions Current Semester 8 / 12
Direct Image of a Set: Properties Theorem Let f : A B. Let E and F be subsets of A. The following is true: 1 If E F then f (E) f (F ) 2 f (E F ) f (E) f (F ) 3 f (E F ) = f (E) f (F ) 4 f (E \ F ) f (E) Philippe B. Laval (KSU) Functions Current Semester 9 / 12
Inverse Image of a Set: s and Examples Let f : A B and let G be a set such that G B. The inverse image of G, denoted f 1 (G) is defined by: Example f 1 (G) = {x A : f (x) G} Remark The above definition does not require that f be injective or have an inverse. f 1 (G) is simply the notation for the inverse image of G. The reader should never think we are talking about the inverse of f. Remark It should be clear to the reader that f 1 (G) A. Consider f : N N defined by f (n) = n 2 1. Let G = {3, 4, 5, 6, 7, 8}, find f 1 (G). Philippe B. Laval (KSU) Functions Current Semester 10 / 12
Inverse Image of a Set: Properties Theorem Let f : A B. Let G and H be subsets of B. The following is true: 1 If G H then f 1 (G) f 1 (H) 2 f 1 (G H) = f 1 (G) f 1 (H) 3 f 1 (G H) = f 1 (G) f 1 (H) 4 f 1 (G \ H) = f 1 (G) \ f 1 (H) Philippe B. Laval (KSU) Functions Current Semester 11 / 12
Exercises See the problems at the end of my notes on functions. Philippe B. Laval (KSU) Functions Current Semester 12 / 12