Is the set #{1, 1}# closed under multiplication and/or addition?
1 Answer
Explanation:
Here's a multiplication table for
#underline(color(white)(0+0$)color(white)(0+)1color(white)(0)color(white)(0)1color(white)(0))#
#color(white)(0+)1color(white)(0)color(white)(0+)1color(white)(0)color(white)(0)1color(white)(0)#
#color(white)(0)1color(white)(0)color(white)(0)1color(white)(0)color(white)(0+)1color(white)(0)#
Regardless of which element we multiply by which, we get an element of the set. So
The same cannot be said of addition, since
Footnote
Closure is one of the axioms of a group:
A group is a set

Closure: If
#a, b in S# then#a@b in S# 
Identity: There is an element
#I in S# such that#a@I = I@a = a# for any#a in S# 
Inverse: For any
#a in S# there is an element#b in S# such that#a@b = b@a = I#
A commutative group also satisfies:
 Commutativity:
#a@b = b@a# for all#a, b in S#
The set
The normal name of this particular group is