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MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Julia Kim - - PowerPoint PPT Presentation
MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Julia Kim - - PowerPoint PPT Presentation
MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Julia Kim September 11th, 2020 Another warm-up problem What if we had N horizontal lines (including the base)? N l i n e s Intervals What are the following
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Intervals
What are the following sets?
1 [2, 4] ∪ (2, 5) 2 [2, 4] ∩ (2, 5) 3 [π, e] 4 [0, 0] 5 (0, 0)
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Similar sets
What are the following sets?
1 A = {x ∈ Z : x2 < 6} 2 B = {x ∈ N : x2 < 6} 3 C = {x ∈ R : x2 < 6}
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Similar sets
What are the following sets?
1 A = {x ∈ Z : x2 < 6} 2 B = {x ∈ N : x2 < 6} 3 C = {x ∈ R : x2 < 6}
Can we list each set’s elements?
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An engineer, a physicist, and a mathematician are on a trip in Scotland. At some point, they see a black sheep. The engineer exclaims: "In Scotland, sheep are black!" The physicist replies: "In Scotland, some sheep are black." The mathematician says: "In Scotland, there is at least one sheep that is black on at least one side."
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An engineer, a physicist, and a mathematician are on a trip in Scotland. At some point, they see a black sheep. The engineer exclaims: "In Scotland, sheep are black!" The physicist replies: "In Scotland, some sheep are black." The mathematician says: "In Scotland, there is at least one sheep that is black on at least one side." Call S the set of all sheep in Scotland and B the set of all black sheep. Whose sentence is equivalent to which of the following?
1 ∃s ∈ S such that s ∈ B. 2 ∃s ∈ S such that s /
∈ B.
3 ∀s ∈ S, s ∈ B.
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An engineer, a physicist, and a mathematician are on a trip in Scotland. At some point, they see a black sheep. The engineer exclaims: "In Scotland, sheep are black!" The physicist replies: "In Scotland, some sheep are black." The mathematician says: "In Scotland, there is at least one sheep that is black on at least one side." Call S the set of all sheep in Scotland and B the set of all black sheep. Whose sentence is equivalent to which of the following?
1 ∃s ∈ S such that s ∈ B. 2 ∃s ∈ S such that s /
∈ B.
3 ∀s ∈ S, s ∈ B. 4 S ⊆ B. 5 S B. 6 S ∩ B = ∅.
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Describing a new set
An irrational number is a number that is real but not rational.
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Describing a new set
An irrational number is a number that is real but not rational. C is the set of positive, rational numbers and negative, irrational numbers. Write a definition for C using only mathematical notation. (You may use the words “and", “or", and “such that".)
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Describing a new set
An irrational number is a number that is real but not rational. C is the set of positive, rational numbers and negative, irrational numbers. Write a definition for C using only mathematical notation. (You may use the words “and", “or", and “such that".) A =
- x ∈ R
- x2 is rational
- B =
- x ∈ R
- x3 is irrational
- A ∩ B = { x ∈ R | ??? }
A ∪ B = { x ∈ R | ??? }
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Sets and quantifiers
What are the following sets?
1 A = {x ∈ R : ∀y ∈ [0, 1], x < y} 2 B = {x ∈ R : ∃y ∈ [0, 1] s.t. x < y} 3 C = {x ∈ [0, 1] : ∀y ∈ [0, 1], x < y} 4 D = {x ∈ [0, 1] : ∃y ∈ [0, 1] s.t. x < y} 5 E = {x ∈ [0, 1] : ∃y ∈ R s.t. x < y} 6 F = {x ∈ [0, 1] : y ∈ R, x < y}
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Before next class...
- Watch videos 1.4, 1.5, and 1.6
- Download the next class’s slides (no need to look at them!)