Set Operations in Mathematics

Set Operations

• Union: The result of combining two sets to form a new set containing all elements from both sets.
  • Notation: A ∪ B
  • Example: {1, 2} ∪ {2, 3} = {1, 2, 3}
• Intersection: The result of combining two sets to form a new set containing only elements common to both sets.
  • Notation: A ∩ B
  • Example: {1, 2} ∩ {2, 3} = {2}
• Difference: The result of removing all elements of one set from another set.
  • Notation: A \ B or A - B
  • Example: {1, 2, 3} \ {2, 3} = {1}
• Complement: The result of taking all elements not in a given set.
  • Notation: A'
  • Example: If U = {1, 2, 3, 4} and A = {1, 2}, then A' = {3, 4}

Logical Operations

• Conjunction (AND): A binary operation that results in TRUE if both operands are TRUE.
  • Notation: ∧
  • Example: TRUE ∧ TRUE = TRUE, TRUE ∧ FALSE = FALSE
• Disjunction (OR): A binary operation that results in TRUE if at least one operand is TRUE.
  • Notation: ∨
  • Example: TRUE ∨ TRUE = TRUE, TRUE ∨ FALSE = TRUE
• Negation (NOT): A unary operation that results in the opposite of the operand.
  • Notation: ¬
  • Example: ¬TRUE = FALSE, ¬FALSE = TRUE

Properties of Binary Operations

• Commutativity: If the order of the operands does not change the result.
  • Example: a + b = b + a (addition is commutative)
• Associativity: If the order in which operations are performed does not change the result.
  • Example: (a + b) + c = a + (b + c) (addition is associative)
• Distributivity: If an operation can be performed on each operand separately and the results combined.
  • Example: a + (b × c) = (a + b) × (a + c) (addition distributes over multiplication)
• Identity Element: An element that does not change the result when combined with another element.
  • Example: 0 is the identity element for addition, since a + 0 = a
• Inverse Element: An element that, when combined with another element, results in the identity element.
  • Example: -a is the inverse element of a for addition, since a + (-a) = 0

Set Operations

• Union combines two sets to form a new set containing all elements from both sets, denoted as A ∪ B.
• Example: {1, 2} ∪ {2, 3} results in {1, 2, 3}.
• Intersection combines two sets to form a new set containing only elements common to both sets, denoted as A ∩ B.
• Example: {1, 2} ∩ {2, 3} results in {2}.
• Difference removes all elements of one set from another set, denoted as A \ B or A - B.
• Example: {1, 2, 3} \ {2, 3} results in {1}.
• Complement takes all elements not in a given set, denoted as A'.
• Example: If U = {1, 2, 3, 4} and A = {1, 2}, then A' = {3, 4}.

Logical Operations

• Conjunction (AND) is a binary operation that results in TRUE if both operands are TRUE, denoted as ∧.
• Example: TRUE ∧ TRUE = TRUE, TRUE ∧ FALSE = FALSE.
• Disjunction (OR) is a binary operation that results in TRUE if at least one operand is TRUE, denoted as ∨.
• Example: TRUE ∨ TRUE = TRUE, TRUE ∨ FALSE = TRUE.
• Negation (NOT) is a unary operation that results in the opposite of the operand, denoted as ¬.
• Example: ¬TRUE = FALSE, ¬FALSE = TRUE.

Properties of Binary Operations

• Commutativity means the order of the operands does not change the result, e.g., a + b = b + a (addition is commutative).
• Associativity means the order in which operations are performed does not change the result, e.g., (a + b) + c = a + (b + c) (addition is associative).
• Distributivity means an operation can be performed on each operand separately and the results combined, e.g., a + (b × c) = (a + b) × (a + c) (addition distributes over multiplication).
• Identity Element is an element that does not change the result when combined with another element, e.g., 0 is the identity element for addition, since a + 0 = a.
• Inverse Element is an element that, when combined with another element, results in the identity element, e.g., -a is the inverse element of a for addition, since a + (-a) = 0.