Let f: Z-Z be defined as 1x)-{xxis even x+5 if xis odd Show that is a one-to-one correspondence. To show is a one-to-one correspondence we need to show that fis one-to-one and onto. First show that fis one-to-one. Notice that if f(x) is even, then x must be -Select-. Also, if f(x) is odd, then x must be -Select-. This is because in both cases of the function definition for f f(x) differs from x by an -Select- integer. An odd integer plus an odd integer is-Select- while an odd integer plus an even integer is-Select--. Now show that f(a) f(b) implies a - b for both cases. Suppose f(a) f(b) is odd. Then a and bare -Select--. So . Similarly if f(a) f(b) is even then a and bare -Select-. Substituting a and b into f we have the equation a-7- which simplifies to a - which simplifies to ab. Therefore fis one-to-one. To show that fis onto we need to show that for some arbitrary y E Z that there is an x EZ such that f(x)= y. Let yE Z be odd. Then y + 7 is-Select-- f(y + 7) = Similarly, suppose y EZ is even. Then y- 5 is-Select-. fly-5) = Therefore fis both onto and one-to-one and has a one-to-one correspondence.

Let f: Z-Z be defined as 1x)-{xxis even x+5 if xis odd Show that is a one-to-one correspondence. To show is a one-to-one correspondence we need to show that fis one-to-one and onto. First show that fis one-to-one. Notice that if f(x) is even, then x must be -Select-. Also, if f(x) is odd, then x must be -Select-. This is because in both cases of the function definition for f f(x) differs from x by an -Select- integer. An odd integer plus an odd integer is-Select- while an odd integer plus an even integer is-Select--. Now show that f(a) f(b) implies a - b for both cases. Suppose f(a) f(b) is odd. Then a and bare -Select--. So . Similarly if f(a) f(b) is even then a and bare -Select-. Substituting a and b into f we have the equation a-7- which simplifies to a - which simplifies to ab. Therefore fis one-to-one. To show that fis onto we need to show that for some arbitrary y E Z that there is an x EZ such that f(x)= y. Let yE Z be odd. Then y + 7 is-Select-- f(y + 7) = Similarly, suppose y EZ is even. Then y- 5 is-Select-. fly-5) = Therefore fis both onto and one-to-one and has a one-to-one correspondence.

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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