4.6: Summary

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Algebra

The algebra of a given object has rules and procedures specific to that object - the operations and rules of sets aren't exactly the same as the ones of numbers or statements. Taking mathematics in primary and secondary education, we are familiar with the algebra of numbers. Now we can apply those same thought processes to other areas of mathematics to prove statements or identities, and determine solutions for abstract problems.

Example \(\PageIndex{1}\):

An "object" might be said to be "all numbers", or "the Universal set containing all numbers." In algebra, we know that we can do operations to numbers: addition, subtraction, multiplication, and division, among others.

These operations are closed to numbers: when we do addition with numbers, we will always receive a number as the result.

This is true of other objects also: statements, sets, and geometric shapes are all objects with their own distinct operations and properties.

In mathematics, we view algebra as the study of relationships between objects, as well as the study of groups of objects. We are already familiar with some relationships between specific quantities.

Example \(\PageIndex{2}\):

In physics:

\( F = ma\), where \(F\) is force, in newtons, \(m\) is mass, in kg, and \(a\) is acceleration
\(v = \displaystyle \frac{d}{t} \)

We also know that these formulas can be manipulated algebraically - that is, that the formula defines the relationship between the elements of the formula, and can be used to determine any of the elements as required.

Some Objects, Operations, and Properties

Objects

Numbers

Statements

Sets

Geometric shapes

Operations

\(+,-,\cdot, \div\)

\(\neg,\vee,\wedge, \rightarrow \)

\(\subset, \subseteq, ^c, \cup, \cap \)

Reflection, rotation, translation

Properties

Closed

Distributive

Closed

Distributive

Closed

Distributive

Closed

Distributive

Closed operations are those that generate an answer in the same group as the elements operated upon.

Distributive operations are those that, in a mathematical sentence, yield the same result when FOILed into a bracket as when not.

Example \(\PageIndex{3}\):

\(\neg \left( A \vee B\right) \) = \(\neg A \wedge \neg B\)

\(2 \left( 3 + 6 \right) = 2 \left( 3 \right) + 2 \left( 6 \right) \)

The Algebra of Geometry

In geometry, our operations are specific to the type of geometry we are doing. This means that the operations we use in 2-dimensional Euclidean geometry might not necessarily be the same as those we apply in 3-dimensional Euclidean or non-Euclidean geometry. For us, we will consider the 2-dimensional transformations as operations. Translations, reflections, and rotations follow the same rules as other operations: they are closed (when you move a shape, the result is a shape) as well as distributive (the order in which you move, rotate, or reflect a shape has no effect on the result).