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Boolean algebra is a subsection of algebra used for analyzing and simplifying logical statements and binary variables. It operates on binary values where variables are assigned the value of either 0 (false) or 1 (true). The primary operations in Boolean algebra are AND (conjunction, \(\cdot\) or \(\cap\)), OR (disjunction, \(+\) or \(\cup\)), and NOT (negation, represented by a bar over the variable or \(\prime\)). These operations correspond to logical connectives and have properties similar to algebraic systems, such as commutativity, associativity, and distributivity, which enable us to manipulate and simplify expressions. \(f(x, y) = (x + y)xy'\) involves these Boolean operations and showcases how to render complex logical statements into a more manageable form.

The distributive law is a principle that comes into play when simplifying Boolean expressions. It signifies that the OR operation is distributive over the AND operation. This means you can expand expression \(ABC + ABD\) to \(AB(C+D)\) or vice versa. When we have a term like \( (x + y)xy' \) in Boolean Algebra, we can apply this law to distribute \( x \) across the terms inside the parentheses. The simplification process of our example, \( f(x, y) \) relies heavily on applying the distributive law to break down complex terms into a format that can be further simplified using additional Boolean laws.

The consensus theorem is a handy shortcut in Boolean Algebra that helps us to eliminate redundancies in logical expressions without going through the more cumbersome method of applying other Boolean laws repeatedly. The theorem states that \( AB + A'BC = AB + AC \) and its dual \( A + B(A' + C) = A + BC \) summarize the consensus of the terms. In essence, the consensus theorem recognizes that \( B \) and \( B' \) can be 'consensus-ed' out, as it has a presence in both the terms related through \( A \) and \( A' \) and doesn't change the outcome of the expression. This theorem allows us to streamline \( f(x, y) \) from the exercise by eliminating the redundant terms and obtaining a minimal expression. This not only simplifies calculations but also has real-world benefits in optimizing digital logic circuit designs.

A Sum of Products (SOP) form, also known in our case as Disjunctive Normal Form (DNF), is a manner of structuring Boolean expressions as a series of AND operations (products), summed by OR operations. In other words, it's a sum of several product terms and each product term typically contains all Boolean variables in the expression, either in its original or complemented form. For instance, the DNF for a Boolean function \( f \) can be written in the form \( f = A'B + ABC' + AC'D \) and so on. This form is incredibly useful for creating truth tables and implementing logic in digital circuits using AND and OR gates. After applying Boolean laws and distributive properties in the given exercise, the expression \( f(x, y) = xy + xy' \) stands as the DNF of the original function, featuring each variable in its original or complemented form within the product terms.