Binary operations

Question 1

What is a binary operation?

Answer 1

Any mathematical procedure that has two inputs and one output.

Question 2

What does it mean if a binary operation is closed?

Answer 2

All the outputs of the operation belong to the same set as the inputs of the operation.

Question 3

What does it mean if a binary operation * is commutative?

Answer 3

If xy = yx for all x and y

e.g: addition and multiplication are commutative, subtraction and division are not.

Question 4

What does it mean if a binary operation * is associative?

Answer 4

x(yz) = (xy)z for all x, y and z.

e.g: matrix addition is associative, matrix multiplication is not.

Question 5

In a cayley table for the operation *, what does the entry in row A column B denote?

Answer 5

A*B

rows come first!

Question 6

What does it mean if a cayley table is symmetrical about the lead diagonal?

Answer 6

The operation is commutative.

Question 7

What is the identity element for a binary operation?

Answer 7

An element that leaves the other input unchanged, often denoted e.
e.g: the identity for addition is 0, the identity for multiplication is 1.

Question 8

What is the inverse of an element under a binary operation *?

Answer 8

The element * its inverse gives the identity as the output.

e.g: under the addition of real numbers, the inverse of x is -x.