binary operation
m.n=m2+n is associative or commutative?
well, try it and see:
(a.b).c = (a^2+b).c = (a^2+b)^2 + c
a.(b.c) = a.(b^2+c) = a^2+(b^2+c)
not the same at all
a.b = a^2+b
b.a = b^2+a
not the same at all
which one is it steve
associative
To determine whether the binary operation m.n = m^2 + n is associative or commutative, let's first understand what these terms mean in the context of binary operations:
1. Associative: A binary operation * on a set is associative if (a*b)*c = a*(b*c) for all elements a, b, and c in the set. In simpler terms, changing the grouping of the elements does not affect the final result.
2. Commutative: A binary operation * on a set is commutative if a*b = b*a for all elements a and b in the set. In simpler terms, changing the order of the elements does not affect the final result.
Now, let's apply these definitions to the given binary operation m.n = m^2 + n:
Associative Property:
To check if the operation is associative, we need to validate the equation (a*b)*c = a*(b*c) for all elements a, b, and c in the set. Let's substitute the given binary operation into this equation:
((m^2 + n)*m)*p = (m^2 + n)*(m*p)
Simplifying the left side:
((m^2 + n)*m)*p = (m^3 + mn)*p = m^3p + mnp
Simplifying the right side:
(m^2 + n)*(m*p) = (m^2 + n)*mp = m^3p + mnp
Since the left side and right side are equal, the binary operation m.n = m^2 + n is associative.
Commutative Property:
To check if the operation is commutative, we need to validate the equation a*b = b*a for all elements a and b in the set. Let's substitute the given binary operation into this equation:
m.n = n.m
Simplifying the left side:
m^2 + n ≠ n^2 + m
Simplifying the right side:
n^2 + m ≠ m^2 + n
Since the left side is not equal to the right side, the binary operation m.n = m^2 + n is not commutative.
In conclusion, the binary operation m.n = m^2 + n is associative but not commutative.