__maths__>__Foundation of Algebra with Numerical Arithmetics__>__Numerical 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.

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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.

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