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I'd avoid introducing a new notation for such a limited scope: you won't probably use it elsewhere during the class, and it wouldn't be used elsewhere in the literature. I have three suggestions. The first is a simple variation of Trevor Wilson's answer, which mirrors the first relation sign to keep the bridging elements together: $$R\ni(1,3) \And (3,2) \in ... 6 To explain why any particular ordered pair is in S \circ R, you can just show that it satisfies the definition, which says that (a,c) is in S \circ R if there exists b such that (a,b) \in R and (b,c) \in S. To show this is true, you can just give an example of such b and observe that (a,b) \in R and (b,c) \in S. The two things here are ... 3 Thanks for your feedback on my previous answer, which contained a misunderstanding. Here's a new try. I believe the following is the way to express the thought that you were trying to express as (1,2) + (2,1) \implies (1, 1), using only the notation your book seems to be using: \{(1,2)\}\circ\{(2,1)\}=\{(1,1)\}. That is, your book defines a relation as a ... 3 It is important to become comfortable with disambiguation. For example, in listening we must use context to distinguish between the homophones “reed” and “read” as in “I will read a book tonight.” In reading, we must use context to know when to pronounce “read” with a long e sound as in “reed” or with a short e sound as in “red.” The brain is an amazing ... 5 I would like to comment on the question: For the purpose of teaching, should we normalize writing 1x instead of x? not for attempting an answer but to ask why one should consider it a (not so) reasonable idea. I think that behind such question there is the idea that if I write 1x rather than x then the algorithm for sum would be clearer. I think what's ... 2 I wouldn't introduce the notation 1x, which they won't see on any standardized test or in any higher level class, unless it is necessary for student with an IEP. Instead I suggest the following solution (to be proposed to your CE). You say in the comments that you hadn't taught combining like terms to the class, but they had learned it before. It would ... 6 Clearly, this student did not recall how to add x+2x and -y+y, but he did correctly add 10+5=15 (presumably because the 10 and the 15 were "visible"). After discussing with my cooperating educator, she suggested that I rewrite the original system as$$\begin{cases} \color{red}{1}x-\color{red}{1}y=10 \\ 2x+\color{red}{1}y=5 \end{cases} ...
I think the basic answer is that there are all sorts of things that we could write, but don't. We usually leave off things that are redundant, but we can add them back in when convenient. For instance, instead of $1x$ we could also write $1\times 1\times 1\times x$. Or we could write $x + 0$ or even $0x^3 + 0x^2 + x + 0$. We could also write $0yz + 0y^2 + ... 5 Part of our job is to teach students to read and write mathematics properly. We shouldn't be turning out students who say "derive" when they mean "differentiate," say "cancel" when they mean "eliminate," write$\sin(x)$instead of$\sin x$, or who don't know that$\exp\$ is a notation for the exponential function. ...