Which and that are both used to introduce dependent clauses, but how do we decide which one to use?
Here is an example of the correct use of that:
Tiffany likes shoes that are expensive.The term being modified by the clause "that are expensive" is "shoes." The set of things called shoes includes both expensive and inexpensive shoes, so when we say "that are expensive," we are talking only about a subset of the set of all things called shoes.
Let's let the set of all shoes be represented by a red circle and the set of all expensive items be represented by a blue circle.

Then "shoes that are expensive" is represented by the intersection of the two sets, the purple area in Fig. 1. It is a subsetof the set of all things called shoes. Not all shoes are expensive.
Here is an example of the correct use of which:
Tiffany likes emeralds, which are expensive.
The set of things called emeralds are all expensive, so the clause "which are expensive" talks about the whole set of emeralds. There is no inexpensive subset of emeralds. "Which are expensive" simply gives you additional information about this whole set.
So if we let the set of all emeralds be represented by a red circle and let all expensive things be represented again by a blue circle, we see that the intersection of the two sets, represented by the purple area in Fig. 2, includes the entire set of emeralds.

Here is the rule in terms of sets:
If a clause describes the whole set of the term it modifies, the clause in question should be introduced withwhich and separated by one or two commas from the rest of the sentence. (This is a nonrestrictive clause.) If the clause describes only a subset of the term it modifies, then the clause in question should be introduced by that and should not be separated by commas. (This is a restrictive clause.)
Another helpful way to think about this is to view non-restrictive clauses as parenthetical. So when a sentence still makes sense if you imagine the clause inside parentheses or if you imagine the addition of the words, "by the way," you have a nonrestrictive clause and it should be introduced by which and separated from the rest of the sentence by a comma or pair of commas.
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