Hello everyone, in this video we are going to explore group theory. This area of mathematics studies algebraic structures known as groups. A group is a set equipped with an operation that combines any two of its elements to form a third element, in such a way that four conditions are satisfied: closure, associativity, identity, and invertibility.

Consider a set G = {0,1,2,3} with an operation 'star' defined as a * b = (a + b) mod 4. Let's verify whether the operation 'star' is closed under G. The answer is yes, because the remainder when we divide a + b by 4 is either 0, 1, 2, or 3, which means the operation is closed under G.

Is there an identity under the operation 'star'? If so, what is it? The answer is again yes, because with addition mod 4, the number 0 is the identity element. This means adding by 0 leaves the original number unchanged.

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