We discuss fault equivalence and dominance relations for multiple output combinational circuits. The conventional definition for equivalence says that "Two faults are equivalent if and only if the corresponding faulty circuits have identical output functions". This definition, which is based on indistinguishability of the faults, is extended for multiple output circuits as "Two faults of a Boolean circuit are equivalent if and only if the pair of the output functions is identical at each output of the circuit". This is termed as diagnostic equivalence in this paper. "If all tests that detect a fault also detect another fault, not necessarily on the same output, then the two faults are called detection equivalent". Two detection equivalent faults need not be indistinguishable. The definitions for fault dominance follow on similar lines. A novel algorithm based on redundancy identification has been proposed to find the equivalence and dominance collapsed sets based on diagnostic and detection collapsing. Applying the algorithm to a 4-bit ALU would collapse the total fault set of 502 faults to 253 and 155, respectively, according to diagnostic equivalence and dominance. The collapsed sets have 234 and 92 faults, respectively, for detection equivalence and dominance. In comparison, the traditional structural equivalence and dominance collapsing results in 301 and 248 faults, respectively. Finally, we use library-based functional collapsing in a hierarchical system and find that smaller fault sets are obtained with an order of magnitude reduction in CPU time for very large circuits.