-1

If we perform an operation on ordered pair from set A={1,2,3,4}, we can built a table like this: enter image description here

But, since set A is finite and there is only one element "1", "2" etc., why we can double elements in the table? An operation cannot be perform like this: 1*1, because we have only one "1". In other words, my question is: what we choose from given set A: an ordered pair OR each element of an ordered pair?

Josef Klimuk
  • 187
  • 1
    It seems like you're looking at an operator $\oplus$ that is defined for $a \oplus b$, $a \neq b$, right? Usually operators are not defined this way, for example normal addition and multiplication do accept the same element on both sides. – Matti P. Jan 29 '19 at 09:46
  • Actually I though about :□. First number gets multiplied by itself, plus second number. And I think this kind of table is illegitimate in such case. – Josef Klimuk Jan 29 '19 at 09:49
  • 1
    Well, the symbol doesn't matter here. I just chose a generic symbol, because I don't know what kind of operator you are talking about. – Matti P. Jan 29 '19 at 09:50
  • 2
    Choosing an element from a set does not "consume" it. Following your reasoning to the absurd, after having written the column headers, the set $A$ would be empty. –  Jan 29 '19 at 10:16

1 Answers1

2

An operation on $A$ is a mapping from $A\times A$ to $A$. Since $(1,1)$ is an element of $A\times A$, $1*1$ is defined.

There is nothing in the definition of an operation that would forbid us from "doubling" elements in the table. That, in essence, is the only answer we can give to the question

why we can double elements in the table?

5xum
  • 123,496
  • 6
  • 128
  • 204