Question

Prove that an identity element (if it exists) of any algebraic structure is unique

Answer 1

Answer:

It is the case that if an identity element exists, it is unique: If S S S is a set with a binary operation, and e e e is a left identity and f f f is a right identity, then e = f e=f e=f and there is a unique left identity, right identity, and identity element.

Step-by-step explanation:

Step : 1  An essential mathematical fact is that the identity of any binary operation on a set is always unique; no other element of the set can behave as the identity. As a result, the number system makes sure that zero and one are distinct. Examine the equation axe = a for any element a and x in the group; if there are no such elements, the group is not a group. If you find an element a and x that solves the equation, determine whether that element is the identity element; if not, the group is not a group.

Step : 2 Identity can be defined as the sum of your physical characteristics and defining behaviours. Your name, your eyes' shape and colour, and your fingerprint, for instance, are all aspects of who you are. This combination of traits enables you to be unmistakably and distinctively recognized.

Step : 3  One of the following can be used to demonstrate uniqueness: I Show that x = y by assuming that x, y, and S are such that P(x) P(y) is true. (ii) Construct your argument on the premise that x and y are distinct such that P(x) P(y), and then arrive at a contradiction. We also need to demonstrate that x S such that P(x) is true in order to demonstrate uniqueness and existence.

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