Identity Elements in Binary Operations

The video discusses identity elements in mathematics, starting with familiar examples like the additive identity (0) and multiplicative identity (1) in integers. It extends the concept to any binary operation, explaining how an identity element works within a set. The video provides examples of identity elements in different mathematical contexts and highlights the uniqueness of identity elements with a theorem. An illustrative example is given to demonstrate how to identify an identity element in a set.

What is the additive identity for integers?

• -1

• 0

• 1

• 2

Which element is the multiplicative identity for integers?

• 1

• -1

• 2

• 0

In a binary operation, what condition must an identity element satisfy?

• It must be equal to all other elements.

• It must be greater than all other elements.

• It must be less than all other elements.

• It must leave other elements unchanged when operated with them.

What is the additive identity in Z mod n?

• -1

• n

• 0

• 1

Which element is the multiplicative identity in the natural numbers?

• n

• -1

• 0

• 1

What is a key property of identity elements in a set with a binary operation?

• They are always prime numbers.

• They are always even numbers.

• They are always zero.

• They are unique within the set.

According to the theorem, how many identity elements can a set have with respect to a binary operation?

• One

• Two

• Three

• None

In the example set A, which element is identified as the identity?

• c

• b

• a

• d

What must be verified to confirm an element as the identity in a set?

• It must be the largest element.

• It must be the smallest element.

• It must operate with all elements to leave them unchanged.

• It must be equal to all elements.

In the example, what operation is used to verify the identity element?

• Binary operation

• Multiplication

• Subtraction

• Addition