Number systems

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# Examples

• 1.   The condition ‘is equal to’ is a relation on any set of real numbers because, for any x, y in the set, the statement ‘x is equal to y’ is either definitely true or definitely false. This relation is usually denoted by the symbol =. For this relation, each real number in the set is related only to itself!

• 2.   The condition ‘is less than’ is a relation on any set of real numbers, and we usually denote it by the symbol <. For example, −2 < 1, but 1 −2 and 3 3.

• 3.   The condition ‘is the derivative of’ is a relation on any set of functions. We can define

• For example, let f(x) = x3, g(x) = 3x2 and h(x) = 2ex. Then g f because g is the derivative of f, and h h because h is the derivative of h, but f g because f is not the derivative of g.

• 4.   On , we can define a relation

• that is, z1 is related to z2 if the distance between z1 and z2 in the complex plane is less than or equal to 4. For example, 1 + i 2 − i because

• but 1 + i 3+ 5i because

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