If the TV can handle 1080p, set the computer to 1080p. Some TVs only support 1080p on certain input ports. Check the manual for the correct ports. 1080i is used only for broadcast TV where there isn't enough bandwidth available for 1080p.
If your monitor and TV are both 1920 x 1080 there SHOULD be no loss of resolution, but with many TVs it is not that simple. In a word: OVERSCAN. The TV screen is 1920 x 1080. You feed it a video signal of 1920 x 1080. Many TV makers see this as a problem. If they were to display the 1920 x 1080 pixels on the 1920 x 1080 dots on the screen you would see the entire image. Who would ever want to do that? There is nothing worth watching at the edges of the picture; the middle is the interesting part.
This nonsense started in the early days of TV. TV sets had unregulated power supplies and used vacuum tubes in the scanning circuitry. Changes in line voltage, aging of the tubes and other components (accelerated by heat from the vacuum tubes) could make the picture shrink. To stop this from leaving black areas around the picture they used overscan to make the picture bigger (up to about 20%) than the screen. To allow for this, important content, faces, and titles were kept in the center of the picture.
Later transistorized TV sets with regulated power supplies did not have picture shrinkage problems but kept the overscan anyway, probably so their pictures wouldn't look small compared to other sets. LCD TVs, with no possibility of picture shrinkage, have continued the overscan tradition.
Some TVs have a setting to disable overscan. It may be called pixel-to-pixel or fit-to-screen, and may be available on only some input ports. If you can't disable overscan, in addition to losing picture edges, such as the menu bar and dock, the rest of the picture will be blurred because the enlarged image's pixels will no longer match the dots on the screen, There may be a setting for over or under scan in Displays system preference when the TV is connected, but it is much better to disable it in the TV if possible.