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Trichotomy

Trichotomy

The property of real numbers which guarantees that for any two real numbers a and b, exactly one of the following must be true: a < b, a = b, or a > b.

 

 

See also

Equivalence properties of equality

Key Formula

For all a,b∈R, exactly one holds: a<b,a=b,a>b\text{For all } a, b \in \mathbb{R}, \text{ exactly one holds: } a < b, \quad a = b, \quad a > b
Where:
  • aa = Any real number
  • bb = Any real number
  • R\mathbb{R} = The set of all real numbers

Worked Example

Problem: Given a = 5 and b = 3, determine which of the three trichotomy cases applies.
Step 1: Check whether a < b.
5<3False5 < 3 \quad \text{False}
Step 2: Check whether a = b.
5=3False5 = 3 \quad \text{False}
Step 3: Check whether a > b.
5>3True5 > 3 \quad \text{True}
Answer: Exactly one of the three cases holds: 5 > 3. The trichotomy property guarantees this is the only true relationship.

Why It Matters

Trichotomy is foundational to ordering and comparing real numbers. Without it, you could not reliably sort numbers, solve inequalities, or place values on a number line. It also underpins proof techniques like proof by cases, where you split an argument into the three possible orderings of two quantities.

Common Mistakes

Mistake: Thinking that two of the three cases could be true at the same time (e.g., a ≤ b and a = b both holding as separate cases).
Correction: Trichotomy guarantees exactly one case is true, not "at least one." The relations <, =, and > are mutually exclusive. Statements like a ≤ b combine two cases but do not violate trichotomy — they simply cover more than one possibility at once.

Related Terms