# Disambiguating equality

///////

Like the term ‘add’ (represented by the $\, + \,$ sign), the term ‘equal’ (represented by the $\, = \,$ sign) has many different meanings in mathematics. In fact, the $\, = \,$ sign can stand for at least five different types of equality:

1) Identical (or algebraic) equality:

Examples: $\, 3 + 5 = 8 \,$, and $\, (x + y) (x - y) = x^2 - y^2 \,$.

Notation: This type of equality (identical equality) is often denoted by the symbol $\, \equiv \,$. Using this notation, we write $\, 3 + 5 \equiv 8 \,$, and $\, (x + y) (x - y) \equiv x^2 - y^2 \,$.

IMPORTANT: The second example above is only valid if the algebra is commutative,
since the expansion of the left-hand side gives (using the distributive property twice): $\, (x + y) (x - y) \equiv x(x-y) + y(x-y) \equiv x^2 - xy + yx - y^2$,
which is identically equal to the right-hand side if-and-only-if $\, xy \equiv yx$.

2) Conditional (or equational) equality:

Examples: The values of $\, x \,$ that satisfy the equation $\, 3x^2 - 5x + 2 = 0 \,$,
and the values of $\, x \,$ and $\, y \,$ that satisfy the equation $\, 3x + 5y = 2 \,$.

3) Relational or equivalence equality:

Example: $\, x = y \,$ if-and-only-if $\, x - y \,$ is divisible by $\, 7.$

Notation: This type of equality is often denoted by the symbol $\, \cong \,$.
Using this notation, we can express our example as $\, x \cong y \,$ if $\, x - y \,$ is divisible by $\, 7.$

4) Defining equality:

Example: $\, C \,$ is defined to be equal to $\, A + B \,$.

Notation: $\, C \stackrel {\mathrm{def}}{=} A + B.$

5) Assigned equality:

Example: $\, R \,$ is assigned the value of $\, P + Q \,$.

Notation: Assigned equality is often denoted by the symbol $\, := \,$
and using this notation we can express the example as $\, R := P + Q \,$.

///////