What are basic quantities

Definition set of an equation

In this chapter we look at what the definition set of an equation is.

synonym

  • Domain of an equation

context

Given an equation with one variable z. B. \ (x + 1 = 2 \). Our goal is to find out for which substitutions for \ (x \) a true statement arises. The set of solutions to an equation depends on which values ​​can be used for \ (x \). This leads us to the concept of Definition set, i.e. the set of all permitted substitutions. If no definition set is given, we need to determine the definition set.

The definition set is determined in three simple steps:

1) You allow all numbers that you know as substitutions for \ (x \).

2) Remove all stakes that violate arithmetic rules from this set.

3) From this set you remove all entries that are not meaningful in terms of content.

1) What could I use for \ (x \)?

Basically, you could use any number you know.

The Base amount is the set of all values ​​that could be used in principle.

Put simply, it means the largest set of numbers that you currently know: In school, equation theory is usually limited to the set of real numbers. Unless otherwise specified, \ (\ mathbb {G} = \ mathbb {R} \) applies.

2) What can I use for \ (x \)?

The violation of an establishment against a calculation rule requires a restriction.

The maximum definition set contains all elements of the basic set that can be used without violating the calculation rules.

Examples

  • Equation: \ (x + 1 = 2 \)
    Maximum definition set: \ (\ mathbb {D} _ {\ text {max}} = \ mathbb {R} \)
    (We can substitute all real numbers for \ (x \) without violating a calculation rule.)

  • Equation: \ (\ frac {1} {1-x} = 3 \)
    Maximum definition set: \ (\ mathbb {D} _ {\ text {max}} = \ mathbb {R} \ setminus \ {1 \} \)
    (If we substitute for \ (x = 1 \), the denominator of the fraction becomes zero. However, division by zero is not permitted. We must therefore exclude the \ (1 \) from the definition set.)

annotation

\ (\ mathbb {D} _ {\ text {max}} = \ mathbb {R} \ setminus \ {1 \} \) we say "D max is R without 1" and simply means that the maximum set of definitions is the set of real numbers minus 1. This \ (\ mathbb {R} \ setminus \ {1 \} \) ("R without 1") is mathematically a difference set.

3) What can I use for \ (x \)?

The meaning of the content of the variable may require a (further) restriction.

The Definition set contains all elements of the maximum definition set that are useful as substitutions within the scope of the task.

Examples

  • Equation: \ (x + 1 = 2 \)
    Maximum definition set: \ (\ mathbb {D} _ {\ text {max}} = \ mathbb {R} \)
    Meaning of the variable of the task: \ (x \) stands for a number of students
    Definition set: \ (\ mathbb {D} = \ mathbb {N} \)
    (We can only use natural numbers for \ (x \): There are no \ (0 {,} 5 \) or \ (\ sqrt {2} \) students!)

  • Equation: \ (x + 1 = 2 \)
    Maximum definition set: \ (\ mathbb {D} _ {\ text {max}} = \ mathbb {R} \)
    Meaning of the variable of the task: \ (x \) stands for the duration of a game
    Definition set: \ (\ mathbb {D} = \ mathbb {R} ^ {+} _ {0} \)
    (We can only use nonnegative real numbers for \ (x \): a duration is never negative!)

annotation

  • In the simplest cases: \ (\ mathbb {G} = \ mathbb {D} _ {\ text {max}} = \ mathbb {D} = \ mathbb {R} \), d. H. we are allowed to substitute all real numbers for \ (x \) without violating a calculation rule or without content-wise meaningful solutions.
  • Unfortunately, not all mathematicians delimit the three terms mentioned above as clearly as we do. In many school books, “basic quantity” is used as a synonym for “definition quantity”: The work order “determine the basic quantity” usually means the same thing as “determine the definition quantity”.
  • If the definition set of an equation is given in the task, it is either the maximum definition set or a definition set restricted for content-related or arbitrary (!) Reasons. The person responsible for the task can restrict the set of definitions as he likes!

outlook

Every number from the definition set which, when substituted for \ (x \), leads to a true statement, is called the solution of the equation. An equation can have no solution, exactly one solution, a finite number of solutions, or an infinite number of solutions. The solutions are summarized in the solution set.

Our goal is to determine the solution set (\ (\ rightarrow \) solve equations).

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