Set Operations in Mathematics

Study Notes

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}

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

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

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}.

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.

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.