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Thought-Process to Discover Knowledge

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mathsFoundation of Algebra with Numerical ArithmeticsNumerical Arithmetics: Laws and Properties for Algebra

### Laws and Properties of Arithmetics : Comparison

This page introduces the Trichotomy property and Transitivity properties of numerical comparison. To understand in-equalities in algebra, these properties are used.

click on the content to continue..

Given two real numbers p and q, which of the following is correct?

• p>q
• p=q
• p<q
• Any one of the above three possibilities
• Any one of the above three possibilities

The answer is "Any one of the above three possibilities".

Trichotomy Property: For any numbers p,q in RR, one and only one of the following is true.
p<q
p=q
p>q
Note: sometimes, p<=q is considered. This specifies, consider pair of numbers p and q such that p is less than or equal to q.

It specifies which numbers to choose. That is, if q is 4, then p can be any negative number or 0, 1, 2, 3, 4. It defines a set of numbers for p.

In trichotomy property, for any two numbers (for example, 4 and 3), one and only one of the three possibilities is true.

Which of the following is a meaning for the word "trichotomy"?

• Division into three parts
• Division into three parts
• trying repeatedly without giving up

The answer is "Division into three parts". tri- means three and -chotomy means division-into-parts

For three real numbers p,q,r, it is given that p<q and q<r, then which of the following is correct?

• p<r
• p<r
• p can be lesser than, equal or greater than r

The answer is "p<r".

Transitivity Property: For three numbers p,q,r in RR,

•  if p<q and q<r, then p<r.

•  if p=q and q=r, then p=r.

•  if p>q and q>r, then p>r. Note 1: If p<q and q>r, then the relation between p and r cannot be ascertained.

Note 2: If p<q and q=r, then q can be replaced by r to get p<r.

Which of the following is a meaning for the word "transitivity"?

• commuting between two states
• some property passes across from one to another
• some property passes across from one to another

The answer is "some property passes across from one to another". trans- means across and -itivity means to-go or to-pass.

LPA - Comparison : Comparison of numbers has Trichotomy and Transitivity properties.

slide-show version coming soon