Stroustrup 比较，勘误表？

Question

In Stroustrup C++ 4th Ed Page 891, where comparisons properties are described. He explains that the function cmp can be represented by less than < for a strict weak ordering. I'm confused by his explanation of "Transitivity of equivalence" as follows;

Transitivity of equivalence: Define equiv(x,y) to be,(cmp(x,y)||cmp(yx)), If equiv(x,y) and equiv(y,z), then equiv(xz), The last rule is the one that allows us to define equality (x==y) as,(cmp(xy)||cmp(y,x)) if we need ==.

Should this instead be defined as follows?

cmp is <= and equiv(x,y) = (cmp(x,y) && cmp(y,x))

Appreciate your guidance.

Answer 1

This is not errata.

equiv(x,y) := !(cmp(x,y)||cmp(y,x))
x := x
y := x

substituting in:
!((x < x) || (x < x))
!((false) || (false))
!(false)
true

Answer 2

Can this instead be defined as follows?

cmp is <= and equiv(x,y) = (cmp(x,y) && cmp(y,x))

Yes, that also gives you consistent definitions.

Should this instead be defined as follows?

It isn't better than the definitions we use, so I'd suggest no , mostly because there's loads of existing code written for the current definition.

Instead of the current definition of Compare

Compare is a set of requirements expected by some of the standard library facilities from the user-provided function object types.

The return value of the function call operation applied to an object of a type satisfying Compare , when contextually converted to bool , yields true if the first argument of the call appears before the second in the strict weak ordering relation induced by this type, and false otherwise.

• For all a , cmp(a,a)==false
• If cmp(a,b)==true then cmp(b,a)==false
• If cmp(a,b)==true and cmp(b,c)==true then cmp(a,c)==true

It would instead be

The return value of the function call operation applied to an object of a type satisfying Compare , when contextually converted to bool , yields false if the first argument of the call appears after the second in the strict weak ordering relation induced by this type, and true otherwise.

• For all a , cmp(a,a)==true
• If cmp(a,b)==false then cmp(b,a)==true
• If cmp(a,b)==true and cmp(b,c)==true then cmp(a,c)==true