I'm currently doing work on discrete mathematics in my free time and am having some difficulties with understanding some questions pertaining to relations and functions. To be specific, I'm stuck on the question:
Why do we take $a, b,$ and $c$ always to be constant and $x ,y ,z$ as variables?
The premise to your question isn't exactly true; like many here, I've seen $a,b,c$ used as variables countless of times. Having said that...
Sometimes $a,b,c$ are used as constants in a text when distinguishing constants from variables in polynomials; for example, when discussing polynomials (for example, the Fundamental Theorem of Algebra), one is likely to see notation $$p(z) = a + c_k(z-z_0)^k+c_{k+1}(z-z_0)^{k+1}+...+c_n(z-z_0^n),$$ where $a,c$ are constants, $p$ is a function, and everything else is a variable. This practice has most likely become convention for clarification; when using a large number of different symbols, it's harder to read $$x_1, x_2, x_3, x_4, x_5, x_6, \text{ where the first three are constants and the last three are variables}$$ than it is for $$a,b,c \text{ are constants, } x,y,z \text{ are variables.}$$ Why one particular group over the other? Probably because of how algebra is generally taught when we are young. You're much more likely to see "$y=mx+b$" or "$y=ax+b$" in middle school rather than "$a=bx+c.$" Nevertheless, that doesn't mean that there's strict convention that $a,b,c$ are always constants or $x,y,z$ are always variables.