What is Identities and Contradictions ?
Look at this equation:
x + 1 = 1 + x
It happens to be true always, no matter what the value of x. (Try it out! What if x is 43?) Every number is a solution. Equations like this are called identities. (Because they are always true, they can also be called laws, or rules).
On the other hand, look at this:
x + 1 = x + 5.
This is a contradiction no matter what the value you choose for x, the equation will be false. The solution set of this equation is the empty set.
How can I recognize an identity or a contradiction?
Example 1: Is this equation an identity, a contradiction, or neither?
x(x + 2) = (x + 1)2 - 1
We'll simplify and try to solve it:
x2 + 2x = (x2 + 2x + 1) -1
x2 + 2x = x2 + 2x.
This is obviously true for all x, so the equation was an identity.
Example 2: Is this equation an identity, a contradiction, or neither?
2(x + 1) = 2x + 1
Let's simplify and try to solve it.
2x + 2 = 2x + 1
2 = 1.
If we get to a point where something is obviously false (like 2 = 1!), we know we have a contradiction.