If S = {0,1,2,3,4} and (a,b) is an arbitrary ordered pair such that a ∈ S and b ∈ S, which of the matchings in Exercises 12-15 are binary operations on S? Construct operation tables.
- (a,b) ----> a * b = 2a - b
- (a,b) ----> a ∇ b is the maximum of a and b
- (a,b) ----> a Δ b = either a or b
- (a,b) ----> a ∬ b = 2
Background: I'm on the introductory Review section of my dad's old "Algebra and Trigonometry" textbook from 1971 (authors: Kane, Oesterle, Bechtel, Finco) and everything described is great, but these symbols looked like they popped up out of nowhere. Can anyone explain what they are and how they are being used? Are they related to a branch of mathematics that is more advanced than algebra?
When the book writes "$a \nabla b$ is the maximum of $a$ and $b$", that is a more concise way of writing "Define $\nabla$ as follows: for every $a,b$, $a \nabla b$ is the maximum of $a$ and $b$". Then, the book is asking you to determine whether the map $(a,b) \mapsto a \nabla b$ is a binary operation on $S$.
In other words, $\nabla$ is not an existing symbol that already has an accepted meaning. Rather, the book is defining a new symbol (just for the purposes of the exercise), and then asking you a question about it.
I suggest you review the definition of "binary operation on $S$" carefully. The book is asking you to understand the abstract definition by applying it to some concrete examples. See if you can think of an example of a set $S$ and an operator that is not a binary operation on $S$.