 When writing about mathematics, we often use mathematical or logical symbols in place of words We should be cautious to not overdo it Because oftentimes we can put too many symbols that even though they're mathematically precise It can make it very confusing to read. And so instead It's often better to use words instead of mathematical symbols when appropriate, right? There's many times where the symbol is just more precise. It's more exact. It's more brief than words can ever do So use mathematical symbols in that setting, but if we can have the same amount of clarity with words Typically, I should say if we can have the same amount of precision the same amount of exactness with words over mathematical symbols We often should use the words instead because it'll make it more clear but oftentimes words like in English can be ambiguous and the And the mathematical symbols that add a precision that words alone cannot do so good mathematical writing is finding the right balance between symbols and words and so In this video, I do want to mention one standard convention because again words versus symbols It's not always there's not always one right way to do it But there is one absolute convention that I do want you to know about and it's written here on the top of the screen in bold Begin each sentence with a word never a mathematical symbol and there's a couple reasons for that one in English Chromatically speaking sentences must begin with a capital letter And if your first symbol is an a like you say a is an element of some capital set a you can't capitalized the symbol Because if you did that you would instead write a is inside of a for which that entirely changes the meaning When beforehand you're just like oh a is an element of the set a you're using a lowercase element to describe the The lowercase letter to describe the element of the capital letter set If you were to capitalize the first letter then you're like a is inside of a this is Russell's paradox That's boom that totally changes the meaning in mathematical context Symbols are case sensitive a lowercase x means something different than an uppercase x like for our course We often use lowercase letters for elements of sets and use capital letters for the sets themselves So we can't just capitalize mathematical symbols that changes their meanings So that's one that's one reason alone why we shouldn't start with mathematical symbols But even if you're like well, I'll just start off with a lowercase level letter potentially. I'm breaking laws of grammar But it also just improves the clarity To write every to start every sentence with a word and by all means that word can sometimes be something like thus comma and thus a is an element of a that's okay It doesn't have to be like a lengthy a lengthy thesis that starts the sentence But a single word can make a big difference. So let's look at a few examples here So one sentence would be a is a subset of B That's a reasonable mathematical sentence But again to avoid starting with a mathematical symbol we could improve it by saying instead The set a is a subset of B. It adds a little bit of clarity and follows the convention right here. So The set is a subset of B another example We have the sentence x is an integer. So 2x plus 5 is an integer We don't want to start with the word x over the symbol x We want to start with a word now with this one We could be a little bit more clever with how we write it instead of saying x is an integer So we could instead say that because x is an integer 2x plus 5 is an integer right notice what happens here We had the word so versus the word because both of these words are suggesting some type of implication This thing happened and therefore this thing happened as a consequence right this then that And the placement of the word so then suggests that this was the primary part of the sentence And then this is the implication that follows now conversely if we move the word because into play here We still have statement a implies B. We've now shifted our words This is now the main part of the sentence and this is just a support Putting that word in there at the very beginning. So just a changing of the phrasing We didn't really even have to add any new words. We just slightly rephrased it to get Not so that the symbol doesn't start the sentence anymore. Okay, so because x is an integer 2x plus 5 is an integer also That's a better way to say that sentence and then as a last example here X squared minus x plus 2 equals 0 has two solutions. This is an equation. It's still mathematical symbols We should not start the sentence that way Capitalizing it doesn't make it any better right because again that changes the meaning capital X squared minus lowercase x plus 2 equals 0 has two solutions That's very very very different. So we can't capitalize to solve this We don't want to start with mathematical symbols instead We could say something like the equation x squared minus x plus 2 equals 0 has two solutions It might feel a little bit more wordy because you're adding more words there But in the in the end this stylistic change will make the math compose so much better because We can start every sentence with them with words not symbols because if we didn't do that we can easily get into traps like Okay, a is inside of a So that's a sentence and then my next sentence is then a b is inside of b And then our next sentence would be something like a is inside of a union b your sentences Could just be mathematical symbols with no words whatsoever. It's like, uh, you know, that's that maybe that's the right thing to say But we can then fix it like so So so that it does believe it make it it makes it compose a little better now Of course starting every sentence with the exact same can then give a not this you can of course Replace it with things like thus but of course more words help a lot We don't want to have too many symbols and that's why we start every sentence with a least one English word and that's gonna bring us to the end of our lecture 16 here. Thanks for watching If you liked these videos or learned anything from them feel free to like them share them with friends who might also want to Learn from these videos subscribe to the channel to see more videos like this in the future And of course if you have any questions, please post them in the comments below and I'll be glad to answer them as soon as I can