Binary Operations

Question 1

What is the closure property in binary operations?

Answer 1

• Definition: A set is closed under a binary operation if performing the operation on any two elements of the set results in an element that is also in the set.
• Example: The set of integers is closed under addition because the sum of any two integers is an integer.

Question 2

What is the commutative property in binary operations?

Answer 2

• Definition: A binary operation is commutative if changing the order of the operands does not change the result.
• Formula: 𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎
• Example: Addition is commutative because 3 + 5 = 5 + 3.

Question 3

What is the associative property in binary operations?

Answer 3

• Definition: A binary operation is associative if the grouping of the operands does not change the result.
• Formula: (𝑎 ∗ 𝑏) ∗ 𝑐 = 𝑎 ∗(𝑏 ∗𝑐)
• Example: Addition is associative because (2 + 3) + 4 = 2 + (3 + 4).

Question 3

What is the distributive property in binary operations?

Answer 3

• Definition: A binary operation is distributive if it distributes over another operation.
• Formula: 𝑎 ∗ (𝑏 + 𝑐) = (𝑎 ∗ 𝑏) + (𝑎 ∗ 𝑐)
• Example: Multiplication is distributive over addition because
  2 × (3 + 4) = (2 × 3) + (2 × 4).

Question 4

What is an identity element in binary operations?

Answer 4

• Definition: An element 𝑒 in a set is an identity element for a binary operation if 𝑎 ∗ 𝑒 = 𝑒 ∗ 𝑎 = 𝑎 for any element
  𝑎 in the set.
• Example: In addition, the identity element is 0 because 𝑎 + 0 = 0 + 𝑎 = 𝑎.

Question 5

What is an inverse element in binary operations?

Answer 5

• Definition: An element 𝑏 is an inverse of 𝑎 under a binary operation if
  𝑎 ∗ 𝑏 = 𝑏 ∗ 𝑎 = the identity element.
• Example: In addition, the inverse of
  𝑎 is −𝑎 because 𝑎 + (−𝑎) = 0.