How do you Prove If a is greater than b and b is greater than c then a is greater than c?

Answer 1

This (If a>b and b>c then a>c) is an example of the transitive law or law of transitivity. A good treatment is in Chapter 1 of the classic textbook A Survey of Modern Algebra by G Birkhoff & S MacLane. My answer is a brief version of theirs. First we establish that we are are working in the type of algebraic structure called an ordered domain. Well-known examples are the integers and the real numbers. We assume we have elements a, b etc. and an operation +, such that a+b is an element of the domain. Fundamental properties (assumed as postulates) include that there is an element called zero (0) such that a+0=0, and that every element a has an inverse x such that a+x=0. The inverse of a is conventionally called -a. Also b+(-a) is conventionally written b-a.
So far this is just a domain. We now define what we mean by an ordered domain. This is a domain in which some of the elements are said to be positive, with the following properties : (the addition principle) if a and b are positive, then a+b is positive; and (the law of trichotomy), for a given a, either a is positive, or a=0, or -a is positive. A relation > is now defined as follows : a>b if and only if a-b is positive.
Having established the definitions and postulates we can now prove the result. If a>b and b>c, then by definition this means that a-b and b-c are positive. Now a-c=a+(-c) and (-b)+b=0, so a-c=a+0+(-c)=a+(-b)+b+(-c)=(a-b)+(b-c), which is the sum of two positive numbers and so is positive by the addition principle. We have proved a-c is positive, i.e. a>c as required.
Note I have also used the associative law a+(b+c)=(a+b)+c which I forgot to mention!
I hope the above isn't too heavy - but if a person asks the original question and doesn't just think "it's obvious" as many people would, I expect that person is probably interested in the fundamentals of mathematics!