Not in their entirety. Some certified Unix systems like Solaris or Mac OS X use open-source software in their core product, or have released portions of their code under an open-source license.
Almost all Unix systems are proprietary; they are not open source, and you usually only get the binary modules, not the source code.
Unix is not open source, it is proprietary. Linux is the open-source version of Unix.
To create a Free OS like the UNIX Operating System, They wanted to create a system that was like UNIX without all of the intellectual property issues that UNIX had. (UNIX was proprietary software).
The main reason is vendor support; as part of the contract the vendor agrees to enhance and fix problems in a timely manner. Also, a number of proprietary UNIX systems are better suited for heavy environments than some of the open source versions. It should be indicated, however, that you can buy a support contract for the open source versions as well. One of the things that determine whether or not to use the open source version versus a proprietary one is what applications you intend to use on the platform. Not all versions of either may support the program you want to use.
The main reason is vendor support; as part of the contract the vendor agrees to enhance and fix problems in a timely manner. Also, a number of proprietary UNIX systems are better suited for heavy environments than some of the open source versions. It should be indicated, however, that you can buy a support contract for the open source versions as well. One of the things that determine whether or not to use the open source version versus a proprietary one is what applications you intend to use on the platform. Not all versions of either may support the program you want to use.
A proprietary version of Unix is also usually tied to the manufacturer's hardware, so they are able to customize it to the chipset. In addition, supposedly the support is better (although some might argue). The proprietary versions have also been around a lot longer than the clones, and are somewhat more mature and stable (again some might argue that).
Linux hosting and Unix hosting are not exactly the same thing. Linux hosting and Unix hosting refer to the software operating system employed by the servers of the host. The primary communication being made when a server is described as Unix or Linux is that it is not a Microsoft product. Unix comes in several proprietary versions, Sun Microsystems (now Oracle) and AT&T are two well known commercial versions of Unix. They own and control intellectual property associated with Unix. Linux is a nonproprietary version of Unix. There are many flavors of this. All are "open source." The intellectual property is controlled by one of the various open source models, GPL being the most well know. If a person says they have a Unix server, they may or may not mean the proprietary version of Unix. For many people, unix is a generic reference and can mean Unix or Linux. If a person says they have a Linux server, then it is a Linux operating system. It may be one of over a hundred different types of Linux, but it is based on an open source version of Unix. Note that it is possible to co-mingle the proprietary and nonproprietary versions of Unix, but this is not typically a matter of concern unless you are an expert or unless you are using Apple's version of "open source" Linux. Apple's version works pretty much like open source Linux, mostly, sometimes.
Freely distributable Unix/Linux systems come with the source code, which can be modified. Proprietary Unix systems are distributed as binary only, with no source code, so you can't modify the system (easily). Other reasons are legal: the proprietary systems are copyrighted, and the freely distributable versions should not have any copywritten code in them, thus allowing them to be distributed without licensing arrangements.
Many versions of Unix can be run on microcomputers. Xenix, Solaris, Mac OS X, and IRIX all run on microcomputers.
No. Licensed Unix implementations are a proprietary flexible general purpose operating system originally marketed for minicomputers and mid-range computers to support multiple users and tasks. When Personal Computers became available, versions of Unix were created for them. Unix was also implemented for the IBM mainframe. This is not the same as Linux which is an open source free system with many of the same capabilities and look & feel of unix but without the licensing requirement of unix.
If you are referring to the desktop as being a graphical user interface then Unix is capable of running without it. A GUI was a more recent addition to Unix, meaning that the earlier versions just used a command line and had no desktop environment at all. Today, you can install versions of Unix or Linux that do not have a graphical component at all, and several administrators (such as myself) prefer not to use a desktop.
proprietary UNIX