Section 4.4 Activity 1 • Give me an example of a relation that is: 1. A bijective function. 2. An injective (one-to-one) function only. 3. A surjective (onto) function only. 4. A function, but neither 1-to-1 or onto. 5. Not a function. Definition We can symbolically define that a function is one-to-one (injective) with: f: XY is one-to-one x1x2 f(x1) = f(x2) x1 = x2 Definition We can symbolically define that a function is onto (surjective) with: f: XY is onto yY xX f(x) = y Image and Inverse Image • If f(x)=y then we say that – The image of x is y – The inverse image (pre-image) of y is x Activity #2 • What is the image of Adams? • What is the image of Epp? • What is the inverse image of B? • What is the inverse image of A? The inverse of a function • The inverse of function f is noted as f-1 f-1 ={ (b,a) : (a,b) f } If f = { (1,square), (2,circle), (3,triangle) } What is f-1 Activity #3 Let • f: X → Y as shown by : • g: RR • h: RR g(x) = 2x + 3 h(x) = x2 + 4 Find the inverse of each of these functions. Is the inverse function really a function? • You identified three inverses in Activity #2. • Which of these were actually inverse functions? What kinds of functions produce inverse functions? • What is the definition of a function? • How does this effect whether or not a function has an inverse? Functions/Inverse Functions in CS • Caesar Cypher c: string string c(string,n) : Consider a=0, b=1, c=2, …. z=25 For each character in the input string replace the character with the character formed by En(x) = (x + n) mod 26 For example, En(“y”) = (24 + 3) % 26 = 1 = “b”