There are three basic properties of equality. These are known as the reflexive, symmetric and transitive properties. Let’s take a look at each of these.
Reflexivity: This means that any object or number is equal to itself. Let’s draw an analogy here: In a perfect world, every individual likes herself. That makes the relationship of ‘liking’ reflexive.
Symmetry: This means that if an object equals a second object, then the second one equals the first one. In a perfect world, if Alice likes Bill, then Bill also likes Alice. That makes ‘liking’ symmetric.
Transitivity: This means that if an object equals a second one, and the second one equals a third one, then the first one equals the third one. If Alice likes Bill, and Bill likes Charlie, then Alice likes Charlie. So ‘liking’ is transitive.
A mathematical statement of equality does not change if either side of the equation is altered using the same mathematical operation. For instance, if a, b and c are real numbers:
If a = b, then a + c = b + c, because you have performed the same operation, that is, the addition of c, on both sides of the equation.
If a = b, then a – c = b – c
If a = b, then a * c = b * c
If a = b and c is non zero, then a / c = b / c
These ideas can be generalized to the following forms as well: If a = b, and c = d, then a + c = b + d, and so on. These follow as a direct corollary from the fact that if two numbers are equal, then either can be substituted for the other one in formulae and equations.
This idea can be extended to the following: If F(x) is any function, and a = b, then F (a) = F (b), provided that each side of the equation makes sense, or is well defined, meaning that it does not involve operations like division by zero that lead to undefined results.
Equality can also be defined for geometric objects, sets, and other mathematical entities. All you need to do is define a relationship between them that satisfies the reflexive, symmetric and transitive properties. For instance, you may say that two circles are ‘equal’ if and only if they have the same radii. Check for yourself that this relationship satisfies the three basic properties of equality. The same is true for the congruence of triangles, that is, if all three sides of a triangle are respectively equal to the three sides of another triangle.
