Set representation and manipulation

U: A universal set is a set containing everything.

Venn diagram: used to visualise the possible relations among a collection of sets.

Example: Venn diagram of $A\ \subseteq\ U$.

Complement of a set: contains all the elements in the universal set U but not in A.

$\overline{A} = U-A$

U = {1, 2, 3, 4, 5}

A = {2, 3}

$\overline{A}$ = {1, 4, 5}

Example exercise

Use a Venn diagram to show that $A\ (+)\ B = A\ \cup B\ -(A\ \cap B)$:

Same for both lol

De Morgan's laws

Mathematical statements and concepts are related thru their opposites -> intersection and union of sets.

• The complement of the union of two sets A and B is equal to the intersection of their complements: $\overline{A \cup B} = \overline{A} \cap \overline{B}$

• The complement of the intersection of two sets A and B is equal to the union of their complements: $\overline{A \cap B} = \overline{A} \cup \overline{B}$

Law of sets: commutative, associative and distributive

• Commutativity: set union, set intersection and symmetric difference. Set difference IS NOT commutative.

• Associativity: grouping of elements, does not affect the result. Set union, set intersection and symmetric difference apply. Set difference IS NOT associative.

• Distributivity: law of multiplication and division.

Commutative laws

The commutative laws state that the order in which two sets are combined does not affect the result.

UNION:

Let $A$ and $B$ be two sets, then:

$A \cup B = B\cup A$

INTERSECTION:

Let $A$ and $B$ be two sets, then:

$A \cap B = B\cap A$

Associative laws

The associative laws state that the grouping of sets does not affect the result when performing union or intersection.

UNION:

Let $A$, $B$ and $C$ be three sets, then:

$(A \cup B) \cup C = A \cup( B\cup C)$

INTERSECTION:

Let $A$, $B$ and $C$ be three sets, then:

$(A \cap B) \cap C = A \cap( B\cap C)$

Distributive laws

The distributive laws describe how union and intersection distribute over each other.

Union over intersection:

Let $A$, $B$ and $C$ be three sets, then:

$A \cup (B \cap C) = (A\cup B) \cap (A \cup C)$

The union of everything in A with everything in common between B and C

equals

The union of everything in common between everything in A and B and everything in A and C

Intersection over union:

Let $A$, $B$ and $C$ be three sets, then:

$A \cap (B \cup C) = (A\cap B) \cup (A \cap C)$

Everything in common between A and the union of everything in B and C

equals

Everything in common between A and B plus everything in common between A and C

Identity laws

The identity laws state that combining a set with the universal set or the empty set yields the set itself.

• Union with empty set: $A \cup \emptyset = A$

• Intersection with universal set: $A \cap U = A$

Complement laws

The complement laws describe the relationship between a set and its complement.

The notations $\overline{A}$ and $A^{c}$ both represent the same concept in set theory and logic:

• Union with complement: $A \cup \overline{A} = A$

• Intersection with complement: $A \cap \overline{A} = \emptyset$

Absorption laws

The absorption laws simplify expressions by showing how combining a set with another set and their union or intersection affects the result.

UNION:

Let $A$ and $B$ be two sets, then:

$A \cup (A \cap B) =A$

INTERSECTION:

Let $A$ and $B$ be two sets, then:

$A \cap (A \cup B) =A$

Double negation laws

The double negation law states that taking the complement of a complement returns the original set: $\left( {\overline{A}} \right)^{c} = A$

Set difference laws

The difference of set A and set B, denoted as A − B or A \ B, is the set of elements that are in A but not in B. This can also be expressed using the intersection with the complement of B:

$A - B = A \cap B^{c}$

For example:

Let A={1,2,3} and B={2,3,4}. Then:

$B^{c} = 1$

$A ∩ B^{c}= \{1,2,3 \}∩\{1\}=\{1\}\ and\ A−B={1}$

Partition of a set

• Disjoint sets: two sets are disjoint if and only if their intersection is $\emptyset$.

A partition of A is a set of subsets of $A_{i}$ of A such as: all the subsets $A_{i}$ are disjoints and the union of all subsets $Ai$ is equal to A.