Identity Element: Examples and Uniqueness Proof

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In this video I introduce the identity element.
Definition: If we have a structure S with binary operation *, then S has an identity element, e, if:
for all x in S, e * x = x * e = x.
Examples)
0 is identity element for the set of integers with addition
1 is the identity element for the set of real numbers with multiplication
the zero matrix is the identity element for the set of all two by two matrices with entries being real numbers along with addition, and the identity matrix is identity element for this set with multiplication.
Important Properties:
1. Identity element is unique
2. Identity element is a structural property
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