Let X = {1,2,3,4,5,6} and let f: X → X and g: X → X be defined by
f(x) = (2x+1) (mod 6)
g(x) = (3x-1) (mod 6)
Find f o g(x): X → X and g o f(x): X → X as subsets of X x X (X cross X).
Here, in this question, how are the given functions pronounced and are they related to modular congruency?
And what does it mean to find the composition in terms of subsets, is the answer itself a set listing notation?
"f o g", or "(f o g)(x)" is properly read as "f composed with g", but is often shortened in practice to something like "f of g", or "f circle g". See
Wikipedia:
The notation g ∘ f is read as "g circle f ", "g round f ", "g about f ", "g composed with f ", "g after f ", "g following f ", "g of f", or "g on f ".
I think I've also heard "f ring g".
Properly speaking, the problem should say "g o f: X → X", not "g o f(x): X → X". The variable x does not belong there. But it is common to bend the notation.
The notation used in your problem, "f(x) = (2x+1) (mod 6)", is a little non-standard; evidently what is meant to say is that you calculate 2x + 1, and then use the number in set X that is congruent to that, modulo 6. I was going to say take the remainder on division by 6, but that could be 0 and can't be 6! If the equal sign is really a congruence sign, then this would be more standard.
Expressing the answer in terms of subsets of X x X means reverting to the definition of a function as a set of ordered pairs. For example, the function f(x) = 7 - x on your set X would be written as {(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)}.
