1. 7.1. (a) Prove, using the definition of concatenation in the picture, that concatenation of strings is associative.

(a) Prove, using the definition of concatenation in the picture, that concatenation of strings is associative. epsilon is a string for each a element sigma, and each string u, ua is also a string (using juxtaposition) From this definition of a string we can inductively define other string operators such as the unary string length operator: |epsilon| = 0 |ua| = |u| + 1, for all strings u and symbols a and the binary concatenation operator, u middot v: u middot epsilon = u u middot (va) = (u middot v) a, for all symbols a Note the semantic distinction made between juxtpostion which defines the string, and concatenation, u middot v, which is what these equations define