What do variables represent, anyway?

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In summary, the statement "let x be a real number. Then x^2+1 does not equal 0" is equivalent to the statement "for all x in lR, x^2+1 does not equal 0". Both statements use the same grammatical structure and have the same meaning in mathematics. The difference lies in the use of quantifiers, where "let x be" implies a singular variable and "for all x" implies a plural variable. For a deeper understanding, one can refer to a book on quantifiers or mathematical logic.
  • #1
HyperbolicMan
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"let x be a real number. Then x^2+1 does not equal 0."

"For all x in lR, x^2+1 does not equal 0"

As far as I know, both of these statements mean exactly the same thing. From a grammatical perspective, in the first statement, x is singular ("a real number"), while in the second, x appears to be plural (preceded by "all" which implies more than one). Is one way 'more correct' than the other, or is there some principle in mathematics that equates the two? Instinctively, I feel like the second is better because it sounds more general, but the first one is easier to visualize.
 
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  • #2
i think both usages are singular. i.e. "everyone here is a nerd." every is singular. it the same as for all x, to me at least.
 
  • #3
your question is not about variables, but about quantifiers. get a book on quantifiers. i.e. propositional calculus, or mathematical logic. or just a book on proofs and read the part on quantifiers. you will never regret it.
 

What do variables represent, anyway?

Variables represent any changing or unknown quantities in a scientific experiment or study. They are used to measure and track the effects of different factors and to determine their relationships with each other.

Why are variables important in scientific research?

Variables are important in scientific research because they allow researchers to manipulate and control certain factors in an experiment to determine their effects on the outcome. They also help to ensure that the results of the study are accurate and reliable.

What are the different types of variables in science?

There are three main types of variables in science: independent, dependent, and controlled. Independent variables are those that are manipulated or changed by the researcher. Dependent variables are the outcome or result of the independent variable. Controlled variables are those that are kept constant throughout the experiment to ensure that they do not affect the results.

How are variables identified and measured in a scientific study?

Variables are identified and measured through careful planning and design of the experiment. Researchers must clearly define and label each variable and determine how it will be measured or quantified. This could involve using instruments, surveys, observations, or other methods of data collection.

Can variables change during a study?

Yes, variables can change during a study. In fact, this is often the purpose of the study - to determine how different variables can affect the outcome. However, it is important for researchers to carefully track and document any changes in variables to ensure the accuracy and validity of the results.

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