Is the set #{1, -1}# closed under multiplication and/or addition?

1 Answer
Jan 15, 2017

Answer:

#{1, -1}# is closed with respect to multiplication, but not addition.

Explanation:

Here's a multiplication table for #{1, -1}#...

#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 #{1, -1}# is closed under multiplication.

The same cannot be said of addition, since #1 + (-1) = 0# is not in the set #{1, -1}#

#color(white)()#
Footnote

Closure is one of the axioms of a group:

A group is a set #S# equipped with a binary operation #@# satisfying the following properties:

  • 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 #S = {1, -1}# with multiplication #@ = xx# satisfies all of these axioms, so is a commutative group.

The normal name of this particular group is #C_2#, the cyclic group of order #2#.