Total order

Key takeaways

Total order belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Total order to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Total order from memory before moving on to harder problems.

Reference excerpt

In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation ≤ {\displaystyle \leq } on some set X {\displaystyle X} , which satisfies the following for all a , b {\displaystyle a,b} and c {\displaystyle c} in X {\displaystyle X} :

a ≤ a {\displaystyle a\leq a} (reflexive). If a ≤ b {\displaystyle a\leq b} and b ≤ c {\displaystyle b\leq c} then a ≤ c {\displaystyle a\leq c} (transitive). If a ≤ b {\displaystyle a\leq b} and b ≤ a {\displaystyle b\leq a} then a = b {\displaystyle a=b} (antisymmetric).

a ≤ b {\displaystyle a\leq b} or b ≤ a {\displaystyle b\leq a} (strongly connected, formerly called totality). Requirements 1. to 3. just make up the definition of a partial order. Reflexivity (1.) already follows from strong connectedness (4.), but is required explicitly by many authors nevertheless, to indicate the kinship to partial orders. Total orders are sometimes also called simple, connex, or full orders. A set equipped with a total order is a totally ordered set; the terms simply ordered set, linearly ordered set, toset and loset are also used. The term chain is sometimes defined as a synonym of totally ordered set, but generally refers to a totally ordered subset of a given partially ordered set. An extension of a given partial order to a total order is called a linear extension of that partial order.

Strict and non-strict total orders For delimitation purposes, a total order as defined above is sometimes called non-strict order. For each (non-strict) total order ≤ {\displaystyle \leq } there is an associated relation < {\displaystyle <} , called the strict total order associated with ≤ {\displaystyle \leq } that can be defined in two equivalent ways:

a < b {\displaystyle a<b} if a ≤ b {\displaystyle a\leq b} and a ≠ b {\displaystyle a\neq b} (reflexive reduction).

a < b {\displaystyle a<b} if not b ≤ a {\displaystyle b\leq a} (i.e., < {\displaystyle <} is the complement of the converse of ≤ {\displaystyle \leq } ). Conversely, the reflexive closure of a strict total order < {\displaystyle <} is a (non-strict) total order. Thus, a strict total order on a set X {\displaystyle X} is a strict partial order on X {\displaystyle X} in which any two distinct elements are comparable. That is, a strict total order is a binary relation < {\displaystyle <} on some set X {\displaystyle X} , which satisfies the following for all a , b {\displaystyle a,b} and c {\displaystyle c} in X {\displaystyle X} :

Not a < a {\displaystyle a<a} (irreflexive). If a < b {\displaystyle a<b} then not b < a {\displaystyle b<a} (asymmetric). If a < b {\displaystyle a<b} and b < c {\displaystyle b<c} then a < c {\displaystyle a<c} (transitive). If a ≠ b {\displaystyle a\neq b} , then a < b {\displaystyle a<b} or b < a {\displaystyle b<a} (connected). Asymmetry follows from transitivity and irreflexivity; moreover, irreflexivity follows from asymmetry.

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Total order

Start with the simplest possible case. Write down what Total order claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Total order before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Total order ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Total order

In research: Total order appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Total order in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Total order is common in secondary-school and first-year university syllabi. It links to neighbouring topics Order theory, Properties of binary relations, Set theory, so understanding it makes those chapters shorter.
In everyday life: Look for Total order outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Total order in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Total order means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Total order out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Total order in simple terms?

In mathematics, a total order or linear order is a partial order in which any two elements are comparable. That is, a total order is a binary relation ≤ {\displaystyle \leq } on some set X {\displaystyle X} , which satisfies the following for all a , b {\displaystyle a,b} and c {\displaystyle c} in…

Why does Total order matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Total order?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Total order.