Transitive property

The transitive property is also known as the transitive property of equality. It states that if two values are equal, and either of those two values is equal to a third value, that all the values must be equal. This can be expressed as follows, where a, b, and c, are variables that represent the same number:

If a = b and b = c, then a = c

Example

If a = b, b = c, and c = 2, what are the values of a and b?

c = 2, and b = c, so plugging 2 in for c, we know that b = 2. We can then plug 2 into the first equation, a = b, to find that a = 2 as well. Therefore, a = b = c = 2.

Transitive property applications

The transitive property may be used in a number of different mathematical contexts. One example is algebra. Given two equal expressions, if we also know that one of the expressions is equal to a third expression, it becomes possible to use the other expressions to solve for missing variables, as in the very simple example above.

The transitive property does not necessarily have to use numbers or expressions though, and could be used with other types of objects, like geometric shapes. Say we have a circle, A, that we know is identical to another circle, B. If we are given a third circle, C, then told that this circle is identical to circle B, then we know that the third circle must also be the same as circle A:

Thus, if we knew the radius, or some other measurement of one of the circles, we know the measurements of all the circles.

Transitive property of inequality

The transitive property of inequality is the equivalent of the transitive property of equality for inequalities. It states:

If a < b and b < c, then a < c

This has applications in algebra as well as other areas of mathematics, but also does not necessarily require the use of numbers. For example, if Jenna is taller than Victoria, and Victoria is taller than Kiera, then by the transitive property, we also know that Jenna is taller than Kiera.