Properties of binary operators

Commutativity

A binary operator is commutative if changing the order of the inputs doesn’t change the output.
For example, addition (+) is commutative because a + b = b + a for any numbers a and b.
multiplication (*) is also commutative: a * b = b * a.
However, subtraction (-) and division (/) are not commutative:
- a - b is generally not equal to b - a.
- a / b is generally not equal to b / a.
- Exponentiation (^) - powers - are also not commutative: a^b is generally not equal to b^a.

Associativity

A binary operator is associative if the way in which inputs are grouped does not affect the output.
For example, addition is associative because (a + b) + c = a + (b + c) for any numbers a, b, and c.
multiplication is also associative: (a * b) * c = a * (b * c).
However, subtraction and division are not associative:
- (a - b) - c is generally not equal to a - (b - c).
- (a / b) / c is generally not equal to a / (b / c).
- Exponentiation is also not associative: (a^b)^c is generally not equal to a^(b^c).

Identity elements

If e*a=a*e for ALL values in the set, e is the identity.
For addition, for example, the identity is 0, because 0 + a = a + 0 = a for any number a.
For multiplication, the identity is 1, because 1 * a = a * 1 = a for any number a.

The inverse of the identity element is the identity element itself.

Inverse elements

The inverse element is the
If a*b=b*a=e, a and b are inverse of each other.
For addition, the inverse of a number a is -a, because a + (-a) = (-a) + a = 0.
For multiplication, the inverse of a number a (assuming a \neq 0) is \frac{1}{a}, because a * \frac{1}{a} = \frac{1}{a} * a = 1.

Not all binary operators have an inverse.

flashcards

Question	Answer
What is a commutative binary operator?	A binary operator is commutative if changing the order of the inputs does not change the output, e.g., a + b = b + a for addition.
Name two commutative binary operators and two non-commutative binary operators with examples.	Commutative: addition (a + b = b + a) and multiplication (a * b = b * a). Non-commutative: subtraction (a - b \neq b - a generally) and division (a / b \neq b / a generally).
Is exponentiation (^) commutative? Give an example.	No, exponentiation is not commutative; a^b is generally not equal to b^a.
What is an associative binary operator?	A binary operator is associative if the way inputs are grouped does not affect the output, e.g., (a + b) + c = a + (b + c) for addition.
Name two associative binary operators and two non-associative binary operators.	Associative: addition and multiplication. Non-associative: subtraction and division (e.g., (a - b) - c \neq a - (b - c) generally).
Is exponentiation associative?	No, exponentiation is not associative; (a^b)^c is generally not equal to a^{(b^c)}.
What is an identity element for a binary operator?	If e * a = a * e for all values in the set, then e is the identity element.
Give the identity elements for addition and multiplication.	For addition, the identity is 0 (0 + a = a + 0 = a). For multiplication, the identity is 1 (1 * a = a * 1 = a).
What is the relationship between the identity element and its inverse?	The inverse of the identity element is the identity element itself.
What defines inverse elements for a binary operator?	If a * b = b * a = e (where e is the identity), then a and b are inverses of each other.
Give the inverse elements for addition and multiplication.	For addition, the inverse of a is -a (a + (-a) = 0). For multiplication, the inverse of a (assuming a \neq 0) is \frac{1}{a} (a * \frac{1}{a} = 1).
Do all binary operators have an inverse?	No, not all binary operators have an inverse.

Properties of binary operators

Commutativity

Associativity

Identity elements

Inverse elements

flashcards

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