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1.2 The Language of Sets

1.2 The Language of Sets 1 / 10 If is a , 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-builder notation: the set of elements x in S such that P(x) is true

{ x ∈ S | P(x)}.

The Language of Sets

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

1.2 The Language of Sets 2 / 10 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-builder notation: the set of elements x in S such that P(x) is true

{ x ∈ S | P(x)}.

The Language of Sets

Fact A 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.

1.2 The Language of Sets 2 / 10 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-builder notation: the set of elements x in S such that P(x) is true

{ x ∈ S | P(x)}.

The Language of Sets

Fact A 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.

1.2 The Language of Sets 2 / 10 Another way to specify a set uses what is called the set-builder notation: the set of elements x in S such that P(x) is true

{ x ∈ S | P(x)}.

The Language of Sets

Fact A 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}.

1.2 The Language of Sets 2 / 10 The Language of Sets

Fact A 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-builder notation: 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 Let A = {1, 3, 5}, B = {5, 1, 3}, and C = {1, 1, 5, 3, 5, 3, 3}. What are the 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, and U0.

Examples

Examples

1.2 The Language of Sets 3 / 10 Is {0} = 0? How many elements are in the set {0, {0}}?

For each nonnegative integer n, let Un = {n, −n}. Find U1, U2, and U0.

Examples

Examples Let A = {1, 3, 5}, B = {5, 1, 3}, and C = {1, 1, 5, 3, 5, 3, 3}. What are the elements of A, B, and C? How are A, B, and C related?

1.2 The Language of Sets 3 / 10 How many elements are in the set {0, {0}}?

For each nonnegative integer n, let Un = {n, −n}. Find U1, U2, and U0.

Examples

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

1.2 The Language of Sets 3 / 10 For each nonnegative integer n, let Un = {n, −n}. Find U1, U2, and U0.

Examples

Examples Let A = {1, 3, 5}, B = {5, 1, 3}, and C = {1, 1, 5, 3, 5, 3, 3}. What are the 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}}?

1.2 The Language of Sets 3 / 10 Examples

Examples Let A = {1, 3, 5}, B = {5, 1, 3}, and C = {1, 1, 5, 3, 5, 3, 3}. What are the 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, and U0.

1.2 The Language of Sets 3 / 10 Important Sets

Fact Certain sets of numbers are so frequently referred to that they are given special 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

Fact The real number line is called continuous because it is imagined to have no holes. Because the integers are all separated from each other, the set of integers is called discrete. The name discrete comes from the distinction between continuous and discrete mathematical objects.

1.2 The Language of Sets 5 / 10 Examples: Using the Set-Builder Notation

Example Describe 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 Fact A set A is not a of a set B if there is at least one element of A that is not an element of B. We write A * B in this case.

Subsets

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

1.2 The Language of Sets 7 / 10

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

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

1.2 The Language of Sets 7 / 10 Examples

Example Which 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 Given elements a and b, the (a, b) denotes the consisting of a and b together with the specification that a is the first element 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 and b = 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.

Cartesian Products

Definition

1.2 The Language of Sets 9 / 10 Two ordered pairs (a, b) and (c, d) are equal if, and only if, a = c and b = 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.

Cartesian Products

Definition Given elements a and b, the symbol (a, b) denotes the ordered pair consisting of a and b together with the specification that a is the first element of the pair and b is the second element.

1.2 The Language of Sets 9 / 10 If A and B are sets, we write A × B for the set of ordered pairs (a, b) with a ∈ A and b ∈ B.

Cartesian Products

Definition Given elements a and b, the symbol (a, b) denotes the ordered pair consisting of a and b together with the specification that a is the first element 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 and b = d.

1.2 The Language of Sets 9 / 10 Cartesian Products

Definition Given elements a and b, the symbol (a, b) denotes the ordered pair consisting of a and b together with the specification that a is the first element 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 and b = 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 1 Is (1, 2) = (2, 1)? √ 2 3 1 Is (2, 9 ) = ( 4, 3 )? 3 If A = {1, 2, 3} and B = {a, b, c, d}, find A × B.

Examples

Examples

1.2 The Language of Sets 10 / 10 √ 2 3 1 Is (2, 9 ) = ( 4, 3 )? 3 If A = {1, 2, 3} and B = {a, b, c, d}, find A × B.

Examples

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

1.2 The Language of Sets 10 / 10 3 If A = {1, 2, 3} and B = {a, b, c, d}, find A × B.

Examples

Examples 1 Is (1, 2) = (2, 1)? √ 2 3 1 Is (2, 9 ) = ( 4, 3 )?

1.2 The Language of Sets 10 / 10 Examples

Examples 1 Is (1, 2) = (2, 1)? √ 2 3 1 Is (2, 9 ) = ( 4, 3 )? 3 If A = {1, 2, 3} and B = {a, b, c, d}, find A × B.

1.2 The Language of Sets 10 / 10