Equation Solving - Solution Sets

Solution Sets

If the solution set is empty, then there are no x_i such that the equation

ƒ (x₁,...,x_n) = c,

in which c is a given constant, becomes true.

For example, let us examine a classic one-variable case. Using the squaring function on the integers, that is, the function ƒ whose domain are the integers (the whole numbers) defined by:

ƒ (x) = x2,

consider the equation

ƒ (x) = 2.

Its solution set is {}, the empty set, since 2 is not the square of an integer, so no integer solves this equation. However note that in attempting to find solutions for this equation, if we modify the function's definition – more specifically, the function's domain, we can find solutions to this equation. So, if we were instead to define that the domain of ƒ consists of the real numbers, the equation above has two solutions, and its solution set is

{√2, −√2}.

We have already seen that certain solutions sets can describe surfaces. For example, in studying elementary mathematics, one knows that the solution set of an equation in the form ax + by = c with a, b, and c real-valued constants, with a and b not both equal to zero, forms a line in the vector space R2. However, it may not always be easy to graphically depict solutions sets – for example, the solution set to an equation in the form ax + by + cz + dw = k (with a, b, c, d, and k real-valued constants) is a hyperplane.