Disambiguating equality

This page is a sub-page of our page on Disambiguation.

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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 \, .

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