 In mathematical writing, we have to write lots of sentences where we use words in English, and we also use mathematical symbols to express ideas here. But when you look at symbols like equal, less than or equal to, set containment, maybe an element containment inside of a set, et cetera. When you look at these symbols here, these mathematical symbols are themselves not words, right? They're mathematical symbols. They express an idea much in the same way that a word does, but they're not words. You can't use them interchangeably for words. You can't use things like equal as a verb. That is, the symbol is not a verb. You can, of course, use it in an equation. You can't use these things as prepositions, or nouns, or words. They're mathematical symbols. It would be more appropriate to use the word itself that you're trying to describe as opposed to using this as some type of shorthand. That's the thing that we often do. We often use this symbol, oh, such and such as a subset. No, no, no. Avoid the misuse of symbols. Actually use them in their proper mathematical context, not as a substitution or an abbreviation of some English phrase or words. Let me give you some examples to illustrate the problem we wanna watch out for. So consider this sentence right here. Since the two sets are equal sign, one is a subset of the other. Can a set be an equal sign? That's not what that symbol means. Now clearly what's happening here is the writer who wrote this sentence is using the equal symbol in replace of the word equal itself. This is completely bad form. Instead we should use the word instead of misusing the symbol. We should write it out as since the two sets are equal, one is a subset of the other. That would be a more correct sentence there. Another example. The empty set is a symbol of every other set. Now what is this symbol right here? Well, this symbol is the subset symbol for which you could say that A is a subset of B, like so. But you can't just like drop the word, the sets A and B and be like, oh, this word means subset, no, no, no, no. That is not the word subset. It's a symbol, it's a relation, it needs two things connected to it. This is gibberish, this is nonsense at this moment. Even if we think we can put sense to it, it's completely bad form. So instead of this sentence, we should actually use the word subset. The empty set is a subset of every set. That is the better form. All right, another one that shows up a lot is things like the following. We use an arrow for some type of logical implication. Since A is an integer and X is, sorry, since A is odd and X is odd, this implies that X squared is odd and A squared is odd. This sentence has a lot of issues here. But the one I'm focusing on here is this implication symbol that this implies that. Don't use implication symbols to connect sentences that are logically connected by implication. Now sure, if you're writing something like X squared equals five, therefore this implies that X equals plus or minus five. If you have like a sequence of mathematical statements, then yeah, you can use some type of implication arrow to suggest how you get from one to the other. That can be appropriate, but in an actual sentence, don't use that symbol to connect statements together. Actually use words. So instead of that, we could say something like since A is odd and any odd number squared is odd, A squared is odd. Or you could even add something like we have A squared is odd. You could even pad that a little bit better if you want to. But don't use the mathematical symbol as if it's a word. It should only be used properly in its mathematical context. And so that's a writing tip I wanna give you for mathematical writing here. Don't misuse symbols as if they're words. And also don't put them in other nonsensical situations, but we don't want to abbreviate our English using mathematical symbols. Now that brings us to the end of lecture 19. Thanks for watching. If you learned anything in these videos, please like them, show your appreciation there, share them with friends that might wanna see things like this as well, subscribe to the channel to see more videos like this in the future. And as always, if you have any questions, feel free to post them in the comments below. And I'll be glad to answer them as soon as I can.