# State the trichotomy law formally

• mr_persistance
In summary: This is a good point. I hadn't considered that. Thank you for bringing it up.4) It's fun learn something new.Yes, learning something new is always fun!
mr_persistance
1. For arbitrary real numbers a & b, exactly one of the three relations hold:
a < b, a > b, a = b.

How do I state this more formally while also being correct?2. The attempt at a solution
a, b ∈ ℝ ( (a < b) ⊕ ( a > b ) ⊕ ( a = b) )

From this I made a truth table 2^3 entries long, and what we need is for the solution to only be true when exactly one relation is true and the rest false. The xoring works logically for 7 of the 8 entries, but fails when all three values are true. One solution off the top of my head is to simply add the following snippet ( ∧ ¬( (a < b) ∧ ( a > b ) ∧ ( a = b) )). That seems really ugly huh? But is that what the statement (exactly one of the three relations hold) turns into?

I am a self learner, anyone want to help me improve? Thank you!

Logical Dog
I think, You can also use this equivalent one:
$$(a \wedge \bar{b} \wedge \bar{c} ) \vee (\bar{a} \wedge b \wedge \bar{c}) \vee (\bar{a} \wedge \bar{b} \wedge c )$$

where a,b,c are the three statements.

mr_persistance
Nice BD! I really like this, I feel there's something really beautiful about your formula. Something about the balancing of truthness between all three statements. One being true, the other two are 'weighed' with it, and the other two ones must be the inverse of the true one for the 'weighing' to come out positive. All three are tested together to find the one of three statements that are true, and because of the nature of the test, if any two are true, there's a contradiction. Did you just think of this on the spot?

Logical Dog
mr_persistance said:
Nice BD! I really like this, I feel there's something really beautiful about your formula. Something about the balancing of truthness between all three statements. One being true, the other two are 'weighed' with it, and the other two ones must be the inverse of the true one for the 'weighing' to come out positive. All three are tested together to find the one of three statements that are true, and because of the nature of the test, if any two are true, there's a contradiction. Did you just think of this on the spot?

No, I played around with it on paper, then double checked the logical equivalence on wikipedia so I don't look like an annoying fool! I have bought schaums outlines of logic and its sitting around on my desk , I am just trying to apply what I know so I don't forget in the long run! =) I think logical equivalences are quite interesting and some quite beautiful and unexpected.

mr_persistance said:
1. For arbitrary real numbers a & b, exactly one of the three relations hold:
a < b, a > b, a = b.

How do I state this more formally while also being correct?
As you have it above, it looks perfectly fine. Is there some reason you wanted to represent the trichotomy using symbolic logic?

1) Just a pure joy from symbolic manipulation ( my brain is weird :D )
2) For more complicated statements, human language is ambiguous, so being able to translate a theorem from English to Logic is invaluable for my own type of thinking and understanding.
2a) I don't have to write as much
2b) It's easier to read the argument when it's a concise line of symbols
3) Some books state their model using a combination of set theory and logic instead of english and I want to be prepared
4) It's fun learn something new.
5) Maybe someday I'll have the luxury of exploring math foundations

mr_persistance said:
1) Just a pure joy from symbolic manipulation ( my brain is weird :D )
2) For more complicated statements, human language is ambiguous, so being able to translate a theorem from English to Logic is invaluable for my own type of thinking and understanding.
The three inequalities you started with are symbols, and are completely unambiguous.
mr_persistance said:
2a) I don't have to write as much
Really? This -- ##(a \wedge \bar{b} \wedge \bar{c} ) \vee (\bar{a} \wedge b \wedge \bar{c}) \vee (\bar{a} \wedge \bar{b} \wedge c )## is easier to write than this -- x < y, or x = y, or x > y?

Not to mention that what Bipolar Demon actually wrote was this (unrendered LaTeX):
(a \wedge \bar{b} \wedge \bar{c} ) \vee (\bar{a} \wedge b \wedge \bar{c}) \vee (\bar{a} \wedge \bar{b} \wedge c )
mr_persistance said:
2b) It's easier to read the argument when it's a concise line of symbols
x < y, or x = y, or x > y is pretty easy to read.
mr_persistance said:
3) Some books state their model using a combination of set theory and logic instead of english and I want to be prepared
The problem is that a few people overuse symbolism to the point that it obfuscates the point they are trying to make. If there's a reason for using symbols instead of a prose explanation, fine, but using symbolism for its own sake should be avoided, IMO.
mr_persistance said:
4) It's fun learn something new.
No argument there.
mr_persistance said:
5) Maybe someday I'll have the luxury of exploring math foundations

## 1. What is the trichotomy law?

The trichotomy law is a mathematical principle that states that for any two real numbers, exactly one of the following is true: the first number is less than the second, the first number is equal to the second, or the first number is greater than the second.

## 2. What is the formal statement of the trichotomy law?

The formal statement of the trichotomy law is as follows: For any two real numbers a and b, exactly one of the following holds: a < b, a = b, or a > b.

## 3. What is the significance of the trichotomy law in mathematics?

The trichotomy law is a fundamental principle in mathematics that allows for the comparison and ordering of real numbers. It is a crucial concept in fields such as algebra, calculus, and geometry.

## 4. Can the trichotomy law be applied to other types of numbers?

Yes, the trichotomy law can be extended to other types of numbers, such as complex numbers and rational numbers. However, it may not hold for certain types of non-numeric quantities, such as vectors or matrices.

## 5. How does the trichotomy law relate to the concept of equality?

The trichotomy law states that there can only be one of three possible relationships between two real numbers: less than, equal to, or greater than. This means that equality is a distinct and unique concept, as it is the only way for two numbers to be equivalent under this law.

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