1. We are watching black and white TV.
2. We are watching black-and-white TV.
3. We are watching a black and white TV.
4. we are watching the black and white TV.
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Are they all grammatical?
All are OK, but #2 is the best. Use hyphens like this when you are creating a single adjective out of 2 or more words.
http://grammar.ccc.commnet.edu/grammar/marks/hyphen.htm
See "a" under Hyphens have other uses
Out of the four sentences you provided, options 1, 2, and 3 are grammatically correct, while option 4 contains a minor error. Let's explain the grammatical aspects of each sentence:
1. "We are watching black and white TV."
This sentence is correct. Here, "black and white TV" functions as a noun phrase, which is used to describe the type of TV being watched. As it is a compound noun, there is no need for any additional articles or determiners.
2. "We are watching black-and-white TV."
This sentence is also grammatically correct. Similar to the first sentence, "black-and-white TV" is used as a noun phrase. In this case, the hyphen is added to connect "black" and "white" since they function as a unit modifying the noun phrase "TV."
3. "We are watching a black and white TV."
This sentence is also acceptable. Here, "a black and white TV" is used, indicating that a specific TV of that type is being referred to. The use of the article "a" suggests an indefinite singular TV of that description.
4. "We are watching the black and white TV."
This sentence has a minor grammatical error. Instead of using the definite article "the," it would be more accurate to use "a" since "a black and white TV" implies an indefinite reference. However, if the intention is to refer to a specific TV that is distinct from others, then it would be grammatically correct to use the definite article "the" ('the black and white TV').
In summary, options 1, 2, and 3 are all grammatically correct, while option 4 would be more accurate if "a" were used instead of "the" unless referring to a specific, identifiable TV.