Subsection 3.3.1 Introduction
The truth table is a universal tool for working with Boolean logic expressions. In principle, it’s all we need. In practice, however, it gets cumbersome quickly. We’ve already seen how that can happen even we’re still dealing with trivial ideas. So we’re going to want something else.
In particular, we’re going to want a set of techniques for manipulating logical expressions to make them more useful. But we’ll need to guarantee that the manipulations that we do cannot affect truth value.
Recall that there’s a natural analogy between algebraic (arithmetic) expressions and logical ones. We can continue that analogy here.
So, for example, in algebra we have that these two expressions are equivalent:
\(2x^3 + 17 + 5x^2 2x^3 + 5x^2 + 17 \)
We can transform the first into the second, without changing its value, by exploiting the fact that addition is commutative (i.e., it doesn’t matter what order we do the additions in).
Similarly, in algebra, we have that these two expressions are equivalent:
\((a + b) (c + d) ac + ad + bc + bd \)
This time, we know that we can transform the first into the second, without changing its value, by exploiting a distributivity property: multiplication distributes over addition.
In algebra, we sometimes use these identities to transform an expression into another one that happens to be more useful for some particular purpose. We also use them to simplify expressions.
For example, these two expressions are equivalent:
\((a + b) (c + d) - bc ac + ad + bd \)
We get the second one by first applying the distributive property and then cancelling out the bc and –bc terms.
Exercises Exercises
Exercise Group.
1. Indicate, for each of the following pairs of arithmetic expressions, whether or not they are equivalent: