Rules for Deductive Reasoning – Syllogism in Mathematics

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A less precise, but correct, statement of The Transitive Property of Equality which is quite useful for constructing mathematical models is:

It two expressions represent the same quantity, the two expressions must be equal.

The key to every “word” problem, or “application” problem, or “modeling” problem is this interpretation of the The Transitive Property of Equality.  However, whenever the model is a well established formula it is wise to use the formula (what was a key step in deriving that formula – make a guess).  I will address this topic in detail in future posts.

Example

The number of man days to complete the job is (6)(14)

The number of man days to complete the job is 21x

Therefore (by the Transitive Property) these two expressions must be equal.

This observation yields the model 21x = (6)(14).

 Example

Some real estate will increase in value.

Anything that will increase in value is a good investment.

Therefore, some real estate is a good investment.

Example

Every odd natural number can be written in the form 2k + 1 for some integer k.

23875 is an odd natural number.

Therefore 23875 can be written in the form 2k + 1 for some integer k.

Every beginning algebra textbook presents some form or another the following First Two Properties of Equations.

(1) : If any expression is added to both sides of an equation the resulting equation is equivalent to the original equation.

(2) : If both sides of an equation are multiplied by the same non-zero real number, the resulting equation is equivalent to the original equation.

The solution of every linear equation is now solved with the same  simple application of syllogism.

For example

Every linear equation in one variable can be solved using the first two properties of equations.

3x + 14 = 11x – 9 is a linear equation in one variable.

Therefore 3x + 14 = 11x – 9 can be solved using the first two properties of equations.

Another example

Every linear equation in one variable can be solved using the first two properties of equations.

is a linear equation in one variable.

Therefore   can be solved using the first two properties of equations.

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2 thoughts on “Rules for Deductive Reasoning – Syllogism in Mathematics

  1. I love rules. I am most acceptable to all rules without cost! At what age in general do our young agree that rules are good or beneficial?

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