2.1 Sets & counting

FE Mechanical

2. Probability & statistics

Our FE Mechanical course is currently in development and is a work-in-progress.

Sets & counting

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Set theory provides the mathematical language for counting and probability theory, allowing us to precisely describe events and their relationships.

Definitions

Set: A set is a well-defined collection of distinct objects, called elements or members. Sets are typically denoted by uppercase letters (A, B, C, etc.), while elements are denoted by lowercase letters.

Notation:

$x \in A$ : “ $x$ is an element of set $A$ .”
$x \in / A$ : “ $x$ is not an element of set $A$ .”
$\emptyset$ or ${}$ : the empty set (contains no elements).
$U$ : the universal set (contains all elements under consideration).

Subset: $A \subseteq B$ means every element of $A$ is also in $B$ .
Proper subset: $A \subset B$ means $A \subseteq B$ and $A \neq = B$ .
Equality: $A = B$ means $A \subseteq B$ and $B \subseteq A$ .
Complement: $\overset{ˉ}{A}$ (or $A^{'}$ or $A^{c}$ ) consists of all elements in $U$ that are not in $A$ .

Set operations

The following are common operations performed on sets.

Definitions

Union: $A \cup B = {x : x \in A or x \in B}$ . The union contains all elements that belong to at least one of the sets.
Intersection: $A \cap B = {x : x \in A and x \in B}$ . The intersection contains all elements that belong to both sets.
Disjoint sets: Sets $A$ and $B$ are disjoint if $A \cap B = \emptyset$ . In other words, the sets share no common elements.
Difference: $A - B = {x : x \in A and x \in / B}$ . The difference is the set of elements in set $A$ but without the elements that are also in set $B$ .

Laws of set theory

Identity laws:

$A \cup \emptyset = A$
$A \cap U = A$
$A \cup U = U$
$A \cap \emptyset = \emptyset$

Idempotent laws:

$A \cup A = A$
$A \cap A = A$

Complement laws:

$A \cup \overset{ˉ}{A} = U$
$A \cap \overset{ˉ}{A} = \emptyset$
$\overline{\overset{ˉ}{A}} = A$
$\overset{ˉ}{U} = \emptyset$
$\overset{ˉ}{\emptyset} = U$

Commutative laws:

$A \cup B = B \cup A$
$A \cap B = B \cap A$

Associative laws:

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

Distributive laws:

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

De Morgan’s laws:

$\overline{A \cup B} = \overset{ˉ}{A} \cap \overset{ˉ}{B}$
$\overline{A \cap B} = \overset{ˉ}{A} \cup \overset{ˉ}{B}$

Counting and combinatorics

Combinatorics provides systematic methods for counting the number of ways events can occur, which is essential for calculating probabilities.

The fundamental counting principle

If one event can occur in $m$ ways and a second event can occur in $n$ ways, then the two events together can occur in $m \times n$ ways.

Permutations

A permutation is an ordered arrangement of objects, where order matters.

Permutations of $n$ distinct objects taken $r$ at a time:

$P (n, r) = \frac{n !}{( n - r )!}$

Also written as $_{n} P_{r}$ or $P_{r}^{n}$ .

Special case — permutations of all $n$ objects: $P (n, n) = n!$

Permutations with repetition

When objects are not all distinct, the number of distinguishable permutations of $n$ objects where there are $n_{1}$ of type 1, $n_{2}$ of type 2, …, $n_{k}$ of type $k$ is:

$P (n; n_{1}, n_{2}, \dots, n_{k}) = \frac{n !}{n _{1} ! n _{2} ! \dots n _{k} !}$

where $n = n_{1} + n_{2} + \dots + n_{k}$ .

Combinations

A combination is a selection of objects where order does not matter.

Combinations of $n$ objects taken $r$ at a time:

$C (n, r) = (r n) = \frac{n !}{r ! ( n - r )!} = \frac{P ( n , r )}{r !}$

Also written as $_{n} C_{r}$ , $C_{r}^{n}$ , or $(r n)$ (binomial coefficient).

Properties:

$(r n) = (n - r n)$ (symmetry)
$(0 n) = (n n) = 1$
$(1 n) = (n - 1 n) = n$

Distinguishing permutations from combinations

Use permutations when:

Order matters
Arranging objects in sequence
Assigning positions or ranks

Use combinations when:

Order doesn’t matter
Selecting a subset
Forming groups or committees

Set definitions and notation

Set: collection of distinct elements; notation $x \in A$ , $x \in / A$
Special sets: $\emptyset$ (empty set), $U$ (universal set)
Subset: $A \subseteq B$ (all $A$ in $B$ ), proper subset: $A \subset B$ ( $A \neq = B$ )
Complement: $\overset{ˉ}{A}$ (elements in $U$ not in $A$ )

Set operations

Union: $A \cup B$ (elements in $A$ or $B$ )
Intersection: $A \cap B$ (elements in both $A$ and $B$ )
Disjoint sets: $A \cap B = \emptyset$ (no shared elements)
Difference: $A - B$ (elements in $A$ not in $B$ )

Laws of set theory

Identity: $A \cup \emptyset = A$ , $A \cap U = A$
Idempotent: $A \cup A = A$ , $A \cap A = A$
Complement: $A \cup \overset{ˉ}{A} = U$ , $A \cap \overset{ˉ}{A} = \emptyset$
Commutative: $A \cup B = B \cup A$ , $A \cap B = B \cap A$
Associative: $A \cup (B \cup C) = (A \cup B) \cup C$
Distributive: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
De Morgan’s Laws:
- $\overline{A \cup B} = \overset{ˉ}{A} \cap \overset{ˉ}{B}$
- $\overline{A \cap B} = \overset{ˉ}{A} \cup \overset{ˉ}{B}$

Counting & combinatorics

Fundamental Counting Principle: $m$ ways for event 1, $n$ ways for event 2 $\to m \times n$ total ways
Permutations: ordered arrangements, $P (n, r) = \frac{n !}{( n - r )!}$
- All $n$ objects: $n!$
- With repetition: $P (n; n_{1}, ..., n_{k}) = \frac{n !}{n _{1} ! n _{2} ! \dots n _{k} !}$
Combinations: unordered selections, $C (n, r) = (r n) = \frac{n !}{r ! ( n - r )!}$
- Symmetry: $(r n) = (n - r n)$
- Special cases: $(0 n) = (n n) = 1$ , $(1 n) = n$
Distinguishing:
- Permutations: order matters (arrangements, rankings)
- Combinations: order does not matter (groups, selections)

Next | 2.2 Laws of probability

Sets & counting

Set theory provides the mathematical language for counting and probability theory, allowing us to precisely describe events and their relationships.

Definitions

Set: A set is a well-defined collection of distinct objects, called elements or members. Sets are typically denoted by uppercase letters (A, B, C, etc.), while elements are denoted by lowercase letters.

Notation:

$x \in A$ : “ $x$ is an element of set $A$ .”
$x \in / A$ : “ $x$ is not an element of set $A$ .”
$\emptyset$ or ${}$ : the empty set (contains no elements).
$U$ : the universal set (contains all elements under consideration).

Subset: $A \subseteq B$ means every element of $A$ is also in $B$ .
Proper subset: $A \subset B$ means $A \subseteq B$ and $A \neq = B$ .
Equality: $A = B$ means $A \subseteq B$ and $B \subseteq A$ .
Complement: $\overset{ˉ}{A}$ (or $A^{'}$ or $A^{c}$ ) consists of all elements in $U$ that are not in $A$ .

Set operations

The following are common operations performed on sets.

Definitions

Union: $A \cup B = {x : x \in A or x \in B}$ . The union contains all elements that belong to at least one of the sets.
Intersection: $A \cap B = {x : x \in A and x \in B}$ . The intersection contains all elements that belong to both sets.
Disjoint sets: Sets $A$ and $B$ are disjoint if $A \cap B = \emptyset$ . In other words, the sets share no common elements.
Difference: $A - B = {x : x \in A and x \in / B}$ . The difference is the set of elements in set $A$ but without the elements that are also in set $B$ .

Laws of set theory

Identity laws:

$A \cup \emptyset = A$
$A \cap U = A$
$A \cup U = U$
$A \cap \emptyset = \emptyset$

Idempotent laws:

$A \cup A = A$
$A \cap A = A$

Complement laws:

$A \cup \overset{ˉ}{A} = U$
$A \cap \overset{ˉ}{A} = \emptyset$
$\overline{\overset{ˉ}{A}} = A$
$\overset{ˉ}{U} = \emptyset$
$\overset{ˉ}{\emptyset} = U$

Commutative laws:

$A \cup B = B \cup A$
$A \cap B = B \cap A$

Associative laws:

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

Distributive laws:

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

De Morgan’s laws:

$\overline{A \cup B} = \overset{ˉ}{A} \cap \overset{ˉ}{B}$
$\overline{A \cap B} = \overset{ˉ}{A} \cup \overset{ˉ}{B}$

Counting and combinatorics

Combinatorics provides systematic methods for counting the number of ways events can occur, which is essential for calculating probabilities.

The fundamental counting principle

If one event can occur in $m$ ways and a second event can occur in $n$ ways, then the two events together can occur in $m \times n$ ways.

Permutations

A permutation is an ordered arrangement of objects, where order matters.

Permutations of $n$ distinct objects taken $r$ at a time:

$P (n, r) = \frac{n !}{( n - r )!}$

Also written as $_{n} P_{r}$ or $P_{r}^{n}$ .

Special case — permutations of all $n$ objects: $P (n, n) = n!$

Permutations with repetition

When objects are not all distinct, the number of distinguishable permutations of $n$ objects where there are $n_{1}$ of type 1, $n_{2}$ of type 2, …, $n_{k}$ of type $k$ is:

$P (n; n_{1}, n_{2}, \dots, n_{k}) = \frac{n !}{n _{1} ! n _{2} ! \dots n _{k} !}$

where $n = n_{1} + n_{2} + \dots + n_{k}$ .

Combinations

A combination is a selection of objects where order does not matter.

Combinations of $n$ objects taken $r$ at a time:

$C (n, r) = (r n) = \frac{n !}{r ! ( n - r )!} = \frac{P ( n , r )}{r !}$

Also written as $_{n} C_{r}$ , $C_{r}^{n}$ , or $(r n)$ (binomial coefficient).

Properties:

$(r n) = (n - r n)$ (symmetry)
$(0 n) = (n n) = 1$
$(1 n) = (n - 1 n) = n$

Distinguishing permutations from combinations

Use permutations when:

Order matters
Arranging objects in sequence
Assigning positions or ranks

Use combinations when:

Order doesn’t matter
Selecting a subset
Forming groups or committees

FE Mechanical

2. Probability & statistics

Our FE Mechanical course is currently in development and is a work-in-progress.

Sets & counting

4 min read

Font

Discuss

Feedback

Set theory provides the mathematical language for counting and probability theory, allowing us to precisely describe events and their relationships.

Definitions

Set: A set is a well-defined collection of distinct objects, called elements or members. Sets are typically denoted by uppercase letters (A, B, C, etc.), while elements are denoted by lowercase letters.

Notation:

$x \in A$ : “ $x$ is an element of set $A$ .”
$x \in / A$ : “ $x$ is not an element of set $A$ .”
$\emptyset$ or ${}$ : the empty set (contains no elements).
$U$ : the universal set (contains all elements under consideration).

Subset: $A \subseteq B$ means every element of $A$ is also in $B$ .
Proper subset: $A \subset B$ means $A \subseteq B$ and $A \neq = B$ .
Equality: $A = B$ means $A \subseteq B$ and $B \subseteq A$ .
Complement: $\overset{ˉ}{A}$ (or $A^{'}$ or $A^{c}$ ) consists of all elements in $U$ that are not in $A$ .

Set operations

The following are common operations performed on sets.

Definitions

Union: $A \cup B = {x : x \in A or x \in B}$ . The union contains all elements that belong to at least one of the sets.
Intersection: $A \cap B = {x : x \in A and x \in B}$ . The intersection contains all elements that belong to both sets.
Disjoint sets: Sets $A$ and $B$ are disjoint if $A \cap B = \emptyset$ . In other words, the sets share no common elements.
Difference: $A - B = {x : x \in A and x \in / B}$ . The difference is the set of elements in set $A$ but without the elements that are also in set $B$ .

Laws of set theory

Identity laws:

$A \cup \emptyset = A$
$A \cap U = A$
$A \cup U = U$
$A \cap \emptyset = \emptyset$

Idempotent laws:

$A \cup A = A$
$A \cap A = A$

Complement laws:

$A \cup \overset{ˉ}{A} = U$
$A \cap \overset{ˉ}{A} = \emptyset$
$\overline{\overset{ˉ}{A}} = A$
$\overset{ˉ}{U} = \emptyset$
$\overset{ˉ}{\emptyset} = U$

Commutative laws:

$A \cup B = B \cup A$
$A \cap B = B \cap A$

Associative laws:

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

Distributive laws:

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

De Morgan’s laws:

$\overline{A \cup B} = \overset{ˉ}{A} \cap \overset{ˉ}{B}$
$\overline{A \cap B} = \overset{ˉ}{A} \cup \overset{ˉ}{B}$

Counting and combinatorics

Combinatorics provides systematic methods for counting the number of ways events can occur, which is essential for calculating probabilities.

The fundamental counting principle

If one event can occur in $m$ ways and a second event can occur in $n$ ways, then the two events together can occur in $m \times n$ ways.

Permutations

A permutation is an ordered arrangement of objects, where order matters.

Permutations of $n$ distinct objects taken $r$ at a time:

$P (n, r) = \frac{n !}{( n - r )!}$

Also written as $_{n} P_{r}$ or $P_{r}^{n}$ .

Special case — permutations of all $n$ objects: $P (n, n) = n!$

Permutations with repetition

When objects are not all distinct, the number of distinguishable permutations of $n$ objects where there are $n_{1}$ of type 1, $n_{2}$ of type 2, …, $n_{k}$ of type $k$ is:

$P (n; n_{1}, n_{2}, \dots, n_{k}) = \frac{n !}{n _{1} ! n _{2} ! \dots n _{k} !}$

where $n = n_{1} + n_{2} + \dots + n_{k}$ .

Combinations

A combination is a selection of objects where order does not matter.

Combinations of $n$ objects taken $r$ at a time:

$C (n, r) = (r n) = \frac{n !}{r ! ( n - r )!} = \frac{P ( n , r )}{r !}$

Also written as $_{n} C_{r}$ , $C_{r}^{n}$ , or $(r n)$ (binomial coefficient).

Properties:

$(r n) = (n - r n)$ (symmetry)
$(0 n) = (n n) = 1$
$(1 n) = (n - 1 n) = n$

Distinguishing permutations from combinations

Use permutations when:

Order matters
Arranging objects in sequence
Assigning positions or ranks

Use combinations when:

Order doesn’t matter
Selecting a subset
Forming groups or committees

Set definitions and notation

Set: collection of distinct elements; notation $x \in A$ , $x \in / A$
Special sets: $\emptyset$ (empty set), $U$ (universal set)
Subset: $A \subseteq B$ (all $A$ in $B$ ), proper subset: $A \subset B$ ( $A \neq = B$ )
Complement: $\overset{ˉ}{A}$ (elements in $U$ not in $A$ )

Set operations

Union: $A \cup B$ (elements in $A$ or $B$ )
Intersection: $A \cap B$ (elements in both $A$ and $B$ )
Disjoint sets: $A \cap B = \emptyset$ (no shared elements)
Difference: $A - B$ (elements in $A$ not in $B$ )

Laws of set theory

Identity: $A \cup \emptyset = A$ , $A \cap U = A$
Idempotent: $A \cup A = A$ , $A \cap A = A$
Complement: $A \cup \overset{ˉ}{A} = U$ , $A \cap \overset{ˉ}{A} = \emptyset$
Commutative: $A \cup B = B \cup A$ , $A \cap B = B \cap A$
Associative: $A \cup (B \cup C) = (A \cup B) \cup C$
Distributive: $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$
De Morgan’s Laws:
- $\overline{A \cup B} = \overset{ˉ}{A} \cap \overset{ˉ}{B}$
- $\overline{A \cap B} = \overset{ˉ}{A} \cup \overset{ˉ}{B}$

Counting & combinatorics

Fundamental Counting Principle: $m$ ways for event 1, $n$ ways for event 2 $\to m \times n$ total ways
Permutations: ordered arrangements, $P (n, r) = \frac{n !}{( n - r )!}$
- All $n$ objects: $n!$
- With repetition: $P (n; n_{1}, ..., n_{k}) = \frac{n !}{n _{1} ! n _{2} ! \dots n _{k} !}$
Combinations: unordered selections, $C (n, r) = (r n) = \frac{n !}{r ! ( n - r )!}$
- Symmetry: $(r n) = (n - r n)$
- Special cases: $(0 n) = (n n) = 1$ , $(1 n) = n$
Distinguishing:
- Permutations: order matters (arrangements, rankings)
- Combinations: order does not matter (groups, selections)

Next | 2.2 Laws of probability

Sets & counting

Set theory provides the mathematical language for counting and probability theory, allowing us to precisely describe events and their relationships.

Definitions

Set: A set is a well-defined collection of distinct objects, called elements or members. Sets are typically denoted by uppercase letters (A, B, C, etc.), while elements are denoted by lowercase letters.

Notation:

$x \in A$ : “ $x$ is an element of set $A$ .”
$x \in / A$ : “ $x$ is not an element of set $A$ .”
$\emptyset$ or ${}$ : the empty set (contains no elements).
$U$ : the universal set (contains all elements under consideration).

Subset: $A \subseteq B$ means every element of $A$ is also in $B$ .
Proper subset: $A \subset B$ means $A \subseteq B$ and $A \neq = B$ .
Equality: $A = B$ means $A \subseteq B$ and $B \subseteq A$ .
Complement: $\overset{ˉ}{A}$ (or $A^{'}$ or $A^{c}$ ) consists of all elements in $U$ that are not in $A$ .

Set operations

The following are common operations performed on sets.

Definitions

Union: $A \cup B = {x : x \in A or x \in B}$ . The union contains all elements that belong to at least one of the sets.
Intersection: $A \cap B = {x : x \in A and x \in B}$ . The intersection contains all elements that belong to both sets.
Disjoint sets: Sets $A$ and $B$ are disjoint if $A \cap B = \emptyset$ . In other words, the sets share no common elements.
Difference: $A - B = {x : x \in A and x \in / B}$ . The difference is the set of elements in set $A$ but without the elements that are also in set $B$ .

Laws of set theory

Identity laws:

$A \cup \emptyset = A$
$A \cap U = A$
$A \cup U = U$
$A \cap \emptyset = \emptyset$

Idempotent laws:

$A \cup A = A$
$A \cap A = A$

Complement laws:

$A \cup \overset{ˉ}{A} = U$
$A \cap \overset{ˉ}{A} = \emptyset$
$\overline{\overset{ˉ}{A}} = A$
$\overset{ˉ}{U} = \emptyset$
$\overset{ˉ}{\emptyset} = U$

Commutative laws:

$A \cup B = B \cup A$
$A \cap B = B \cap A$

Associative laws:

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

Distributive laws:

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

De Morgan’s laws:

$\overline{A \cup B} = \overset{ˉ}{A} \cap \overset{ˉ}{B}$
$\overline{A \cap B} = \overset{ˉ}{A} \cup \overset{ˉ}{B}$

Counting and combinatorics

Combinatorics provides systematic methods for counting the number of ways events can occur, which is essential for calculating probabilities.

The fundamental counting principle

If one event can occur in $m$ ways and a second event can occur in $n$ ways, then the two events together can occur in $m \times n$ ways.

Permutations

A permutation is an ordered arrangement of objects, where order matters.

Permutations of $n$ distinct objects taken $r$ at a time:

$P (n, r) = \frac{n !}{( n - r )!}$

Also written as $_{n} P_{r}$ or $P_{r}^{n}$ .

Special case — permutations of all $n$ objects: $P (n, n) = n!$

Permutations with repetition

When objects are not all distinct, the number of distinguishable permutations of $n$ objects where there are $n_{1}$ of type 1, $n_{2}$ of type 2, …, $n_{k}$ of type $k$ is:

$P (n; n_{1}, n_{2}, \dots, n_{k}) = \frac{n !}{n _{1} ! n _{2} ! \dots n _{k} !}$

where $n = n_{1} + n_{2} + \dots + n_{k}$ .

Combinations

A combination is a selection of objects where order does not matter.

Combinations of $n$ objects taken $r$ at a time:

$C (n, r) = (r n) = \frac{n !}{r ! ( n - r )!} = \frac{P ( n , r )}{r !}$

Also written as $_{n} C_{r}$ , $C_{r}^{n}$ , or $(r n)$ (binomial coefficient).

Properties:

$(r n) = (n - r n)$ (symmetry)
$(0 n) = (n n) = 1$
$(1 n) = (n - 1 n) = n$

Distinguishing permutations from combinations

Use permutations when:

Order matters
Arranging objects in sequence
Assigning positions or ranks

Use combinations when:

Order doesn’t matter
Selecting a subset
Forming groups or committees