Nothing is more essential to mathematical thinking than deduction, the art of working out the consequences — the implications — of precise assumptions.
This posting takes a closer look at the mathematical meaning of “implies” through an example in basic algebra. It assumes that you know just a little bit about propositional logic and its “implication” operator, as explained for example in Logic through Pictures.
Let’s look at the statement that
if x is equal to 1 then x squared is also equal to 1.
In mathematical notation we write this as
x = 1 ⇒ x2 = 1
and this is a true mathematical statement just as you would expect. We typically read the arrow here in English as “implies”, reading the whole statement as “x equals 1 implies x squared equals 1”. We are going to look more closely at the precise mathematical meaning of this implication arrow.
The mathematical meaning of a true statement with variables is that any possible value can be used as the value of each variable, giving a true statement in every case. So in this example the statement should be true for x = 0, x = 1, x = 2, and so on. And if we are talking about the real numbers, it should also be true for ½, the square root of 2, and so on.
For each possible value of x we consider the value of x = 1 and the value of x2 = 1. Each of these will be either true or false. The implication arrow is a function that takes two “truth values” (either T or F) as inputs, and gives a truth value as its result.
Since x = 1 ⇒ x2 = 1 is a true statement, its value must be T (true) for all possible values of x. Let’s take a look at the truth table for the ⇒ operator.
P Q P ⇒ Q F F T F T T T F F T T T
Its value is T for all inputs except P = T and Q = F. In our true statement we will see that all three of the cases that give a value of T occur for certain values of x.
Suppose the value of x is 1. Then the expression x = 1 is true and the expression x2 = 1 is also true. The value of T ⇒ T is T, and 1 = 1 ⇒ 12 = 1 is true as expected.
Next suppose x is -1. The expression x = 1 is false, but the expression x2 = 1 is true. The value of F ⇒ T is T, so -1 = 1 ⇒ (-1)2 = 1 is also true.
Now suppose x is 0. Then the expression x = 1 is false and the expression x2 = 1 is also false. The truth value of F ⇒ F is T, and 0 = 1 ⇒ 02 = 1 is true, also as expected. The inputs to “⇒” are false for x = 2, x = 3, x = ½, and all other values of x as well.
As long as there is no value of x that causes x = 1 to be true and x2 = 1 to be false, the implication is true. For this statement there are no such cases, and we see in detail how our original statement is true, looking at all possible cases and using the “truth table” for the “⇒“ operator.