1.2 The Language of Sets

1.2 The Language of Sets 1 / 10

The Language of Sets

FactA set can be viewed, intuitively, as a collection of objects.

If S is a set, the notation x ∈ S means that x is an element of S .The notation x /∈ S means that x is not an element of S .A set can be specified using the set-roster notation: {1, 2, 3, 4} or{0, 1, . . . , 100}.Another way to specify a set uses what is called the set-buildernotation: the set of elements x in S such that P(x) is true

{ x ∈ S |P(x)}.

1.2 The Language of Sets 2 / 10

The Language of Sets

FactA set can be viewed, intuitively, as a collection of objects.If S is a set, the notation x ∈ S means that x is an element of S .

The notation x /∈ S means that x is not an element of S .A set can be specified using the set-roster notation: {1, 2, 3, 4} or{0, 1, . . . , 100}.Another way to specify a set uses what is called the set-buildernotation: the set of elements x in S such that P(x) is true

{ x ∈ S |P(x)}.

1.2 The Language of Sets 2 / 10

The Language of Sets

FactA set can be viewed, intuitively, as a collection of objects.If S is a set, the notation x ∈ S means that x is an element of S .The notation x /∈ S means that x is not an element of S .

A set can be specified using the set-roster notation: {1, 2, 3, 4} or{0, 1, . . . , 100}.Another way to specify a set uses what is called the set-buildernotation: the set of elements x in S such that P(x) is true

{ x ∈ S |P(x)}.

1.2 The Language of Sets 2 / 10

The Language of Sets

FactA set can be viewed, intuitively, as a collection of objects.If S is a set, the notation x ∈ S means that x is an element of S .The notation x /∈ S means that x is not an element of S .A set can be specified using the set-roster notation: {1, 2, 3, 4} or{0, 1, . . . , 100}.

Another way to specify a set uses what is called the set-buildernotation: the set of elements x in S such that P(x) is true

{ x ∈ S |P(x)}.

1.2 The Language of Sets 2 / 10

The Language of Sets

FactA set can be viewed, intuitively, as a collection of objects.If S is a set, the notation x ∈ S means that x is an element of S .The notation x /∈ S means that x is not an element of S .A set can be specified using the set-roster notation: {1, 2, 3, 4} or{0, 1, . . . , 100}.Another way to specify a set uses what is called the set-buildernotation: the set of elements x in S such that P(x) is true

{ x ∈ S |P(x)}.

1.2 The Language of Sets 2 / 10

Examples

Examples

Let A = {1, 3, 5}, B = {5, 1, 3}, and C = {1, 1, 5, 3, 5, 3, 3}. What arethe elements of A, B, and C? How are A, B, and C related?Is {0} = 0?How many elements are in the set {0, {0}}?For each nonnegative integer n, let Un = {n,−n}. Find U1, U2, andU0.

1.2 The Language of Sets 3 / 10

Examples

ExamplesLet A = {1, 3, 5}, B = {5, 1, 3}, and C = {1, 1, 5, 3, 5, 3, 3}. What arethe elements of A, B, and C? How are A, B, and C related?

Is {0} = 0?How many elements are in the set {0, {0}}?For each nonnegative integer n, let Un = {n,−n}. Find U1, U2, andU0.

1.2 The Language of Sets 3 / 10

Examples

ExamplesLet A = {1, 3, 5}, B = {5, 1, 3}, and C = {1, 1, 5, 3, 5, 3, 3}. What arethe elements of A, B, and C? How are A, B, and C related?Is {0} = 0?

How many elements are in the set {0, {0}}?For each nonnegative integer n, let Un = {n,−n}. Find U1, U2, andU0.

1.2 The Language of Sets 3 / 10

Examples

ExamplesLet A = {1, 3, 5}, B = {5, 1, 3}, and C = {1, 1, 5, 3, 5, 3, 3}. What arethe elements of A, B, and C? How are A, B, and C related?Is {0} = 0?How many elements are in the set {0, {0}}?

For each nonnegative integer n, let Un = {n,−n}. Find U1, U2, andU0.

1.2 The Language of Sets 3 / 10

Examples

ExamplesLet A = {1, 3, 5}, B = {5, 1, 3}, and C = {1, 1, 5, 3, 5, 3, 3}. What arethe elements of A, B, and C? How are A, B, and C related?Is {0} = 0?How many elements are in the set {0, {0}}?For each nonnegative integer n, let Un = {n,−n}. Find U1, U2, andU0.

1.2 The Language of Sets 3 / 10

Important Sets

FactCertain sets of numbers are so frequently referred to that they are givenspecial symbolic names.

R is the set of all real numbers.Z is the set of integers.Q is the set of all rational numbers.

1.2 The Language of Sets 4 / 10

Discrete Math vs Continuous Math

FactThe real number line is called continuous because it is imagined tohave no holes.Because the integers are all separated from each other, the set ofintegers is called discrete.The name discrete mathematics comes from the distinction betweencontinuous and discrete mathematical objects.

1.2 The Language of Sets 5 / 10

Examples: Using the Set-Builder Notation

ExampleDescribe each of the following sets:

{x ∈ R | − 1 < x < 6}{x ∈ Z | − 1 < x < 6}{x ∈ Z+ | − 1 < x < 6}

1.2 The Language of Sets 6 / 10

Subsets

DefinitionIf A and B are sets, then A is called a subset of B , written A ⊆ B , if, andonly if, every element of A is also an element of B .

FactA set A is not a subset of a set B if there is at least one element of A thatis not an element of B . We write A * B in this case.

1.2 The Language of Sets 7 / 10

Subsets

DefinitionIf A and B are sets, then A is called a subset of B , written A ⊆ B , if, andonly if, every element of A is also an element of B .

FactA set A is not a subset of a set B if there is at least one element of A thatis not an element of B . We write A * B in this case.

1.2 The Language of Sets 7 / 10

Examples

ExampleWhich of the following are true statements?

1 2 ∈ {1, 2, 3}2 {2} ∈ {1, 2, 3}3 2 ⊆ {1, 2, 3}4 {2} ⊆ {1, 2, 3}5 {2} ⊆ {{1}, {2}}6 {2} ∈ {{1}, {2}}

1.2 The Language of Sets 8 / 10

Cartesian Products

Definition

Given elements a and b, the symbol (a, b) denotes the ordered pairconsisting of a and b together with the specification that a is the firstelement of the pair and b is the second element.Two ordered pairs (a, b) and (c , d) are equal if, and only if, a = c andb = d .If A and B are sets, we write A× B for the set of ordered pairs (a, b)with a ∈ A and b ∈ B .

1.2 The Language of Sets 9 / 10

Cartesian Products

DefinitionGiven elements a and b, the symbol (a, b) denotes the ordered pairconsisting of a and b together with the specification that a is the firstelement of the pair and b is the second element.

Two ordered pairs (a, b) and (c , d) are equal if, and only if, a = c andb = d .If A and B are sets, we write A× B for the set of ordered pairs (a, b)with a ∈ A and b ∈ B .

1.2 The Language of Sets 9 / 10

Cartesian Products

DefinitionGiven elements a and b, the symbol (a, b) denotes the ordered pairconsisting of a and b together with the specification that a is the firstelement of the pair and b is the second element.Two ordered pairs (a, b) and (c , d) are equal if, and only if, a = c andb = d .

If A and B are sets, we write A× B for the set of ordered pairs (a, b)with a ∈ A and b ∈ B .

1.2 The Language of Sets 9 / 10

Cartesian Products

DefinitionGiven elements a and b, the symbol (a, b) denotes the ordered pairconsisting of a and b together with the specification that a is the firstelement of the pair and b is the second element.Two ordered pairs (a, b) and (c , d) are equal if, and only if, a = c andb = d .If A and B are sets, we write A× B for the set of ordered pairs (a, b)with a ∈ A and b ∈ B .

1.2 The Language of Sets 9 / 10

Examples

Examples

1 Is (1, 2) = (2, 1)?2 Is (2, 39) = (

√4, 13)?

3 If A = {1, 2, 3} and B = {a, b, c , d}, find A× B .

1.2 The Language of Sets 10 / 10

Examples

Examples1 Is (1, 2) = (2, 1)?

2 Is (2, 39) = (√4, 13)?

3 If A = {1, 2, 3} and B = {a, b, c , d}, find A× B .

1.2 The Language of Sets 10 / 10

Examples

Examples1 Is (1, 2) = (2, 1)?2 Is (2, 39) = (

√4, 13)?

3 If A = {1, 2, 3} and B = {a, b, c , d}, find A× B .

1.2 The Language of Sets 10 / 10

Examples

Examples1 Is (1, 2) = (2, 1)?2 Is (2, 39) = (

√4, 13)?

3 If A = {1, 2, 3} and B = {a, b, c , d}, find A× B .

1.2 The Language of Sets 10 / 10