A schema is redundant for a sigma if some relation with that heading and satisfying the FDs in sigma has two equal subrows on the attributes of some FD in the closure of sigma. Eg: If X->Y and Y->Z are in sigma but X->Z is not then X->Z is nevertheless in the closure of sigma, so X->Z also has to hold. So if some relation satisfying sigma's FDs has two rows with the same (X,Y), (Y,Z) or (X,Z) value then the schema is redundant. Ie a schema is redundant when some satisfying relation actually exhibits certain (informally) "redundant" subrows per the closure of a sigma.
A schema is value-redundant for a sigma if some relation with that heading and satisfying the FDs in sigma has an element that when given a different value always gives a relation that doesn't satisfy the FDs in sigma. Ie it has an element value that given the rest of the element values must be that value. Eg in any of the above 3 cases of there being equal subrows (ie XY, YZ or XZ), the element in the determined subrow (ie respectively Y, Z or Z) has to have that value given the rest of the element values. Ie a schema is value-redundant when some satisfying relation actually exhibits a certain (informally) "redundant" subrow per a sigma.
Notice that redundancy is in terms of the closure of sigma but value-redundancy is in terms of just sigma.
The text will go on to show that a schema is redundant for a sigma if and only if it is value-redundant. So to determine redundancy, instead of having to (expensively) calculate the closure of sigma we can just use sigma per value-redundancy (in a less expensive way).
