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# Groups of Sets

The binary operation $*$ for combining sets is defined as $A*B =(A\cup B) - (A\cap B)$.

Prove that $G$, consisting of the set of all subsets of a set $S$ (including the empty set and the set $S$ itself), together with the binary operation $*$, forms a group. You may assume that the associative property is satisfied.

Consider the set of all subsets of the natural numbers and solve the equation $\{1,2,4\}*X = \{3,4\}$.

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Age 16 to 18

Challenge Level

The binary operation $*$ for combining sets is defined as $A*B =(A\cup B) - (A\cap B)$.

Prove that $G$, consisting of the set of all subsets of a set $S$ (including the empty set and the set $S$ itself), together with the binary operation $*$, forms a group. You may assume that the associative property is satisfied.

Consider the set of all subsets of the natural numbers and solve the equation $\{1,2,4\}*X = \{3,4\}$.

Learn about the rules for a group and the different groups of 4 elements by doing some simple puzzles.

This article only skims the surface of Galois theory and should probably be accessible to a 17 or 18 year old school student with a strong interest in mathematics.

Explore the properties of some groups such as: The set of all real numbers excluding -1 together with the operation x*y = xy + x + y. Find the identity and the inverse of the element x.