Question

Define a binary composition with example

Answer 1

A binary operation is a calculation that combines the arguments x and y to x y
In mathematics, a binary operation on a set is a calculation that combines two elements of the set (called operands) to produce another element of the set. More formally, a binary operation is an operation of arity two whose two domains and one codomain are the same set. Examples include the familiar elementary arithmetic operations of addition, subtraction, multiplication and division. Other examples are readily found in different areas of mathematics, such as vector addition, matrix multiplication and conjugation in groups.

Answer 2

◾️◾️Binary operation/ composition on a set S if, and only if when,

∀a, b, a ∗ b ∈ S.

◾️◾️Implicit in this definition is the idea that ∗ is closed, or as we often say,
S is closed under ∗.
The idea here is that we can give a value to a ∗ b from the set in which we started.

◾️◾️Example :-
(a, b) ∗ (c, d) = (ac, bc + d) on Z × Z.
Multiplication of n × n matrices with real entries!!

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