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Properties of Inequality

The following are theproperties of inequality for real numbers. They are closely related to theproperties of equality,但也有重要的差异rences.

Note especially that when you multiply or divide both sides of an inequality by a negative number, you must reverse the inequality.

PROPERTIES OF INEQUALITY
Anti reflexive Property

For all real numbers x ,

  • x ≮ x and x ≯ x
Anti symmetry Property

For all real numbers x and y ,

  • if x < y , then y ≮ x .
  • if x > y , then y ≯ x .
Transitive Property

For all real numbers x , y , and z ,

  • if x < y and y < z , then x < z .
  • if x > y and y > z , then x > z .
Addition Property

For all real numbers x , y , and z ,

  • if x < y then x + z < y + z .
Subtraction Property

For all real numbers x , y , and z ,

  • if x < y then x − z < y − z .
Multiplication Property

For all real numbers x , y , and z ,

  • if x < y , then

    • { x z < y z , if z > 0. x z > y z , if z < 0. x z = y z , if z = 0.