Programmable Logic Controllers. Su Chen Jonathon Lin

      Figure 5.1: Commutative AND function

      Figure 5.2: Commutative OR function

       5.2.2Associative Laws

      Associative laws indicate that the grouping of elements can change without changing the result, provided that all of the operations between elements are the same relationship, either all ANDs or ORs. This means that the order of the AND or OR elements does not change the output result (Figure 5.3 and Figure 5.4).

      Figure 5.3: Associative OR function

      Figure 5.4: Associative AND function

       5.2.3Distributive Laws

      The distributive laws allow us to rearrange elements in combinations of AND and OR relationships. Often by rearranging elements, we can combine portions to simplify the circuits. It has two forms: AND-OR and OR-AND. Figure 5.5 shows an example of the distributive law of AND-OR combination, which has the form of:

      A · (B + C) = A · B + A · C

      Figure 5.6 shows an example of using the OR-AND combination of distributive law. The distributive OR-AND function has the following Boolean expression: