What is identities and contradictions, Mathematics

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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.


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