MAT 137 LEC 0601 Instructor: Alessandro Malus TA: Julia Kim - PowerPoint PPT Presentation

MAT 137 — LEC 0601 Instructor: Alessandro Malusà 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 sets? 1 [2 , 4] ∪ (2 , 5) 2 [2 , 4] ∩ (2 , 5) 3 [ π, e ] 4 [0 , 0] 5 (0 , 0)

Similar sets What are the following sets? 1 A = { x ∈ Z : x 2 < 6 } 2 B = { x ∈ N : x 2 < 6 } 3 C = { x ∈ R : x 2 < 6 }

Similar sets What are the following sets? 1 A = { x ∈ Z : x 2 < 6 } 2 B = { x ∈ N : x 2 < 6 } 3 C = { x ∈ R : x 2 < 6 } Can we list each set’s elements?

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."

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 .

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 . 4 S ⊆ B . 2 ∃ s ∈ S such that s / ∈ B . 5 S � B . 3 ∀ s ∈ S , s ∈ B . 6 S ∩ B = ∅ .

Describing a new set An irrational number is a number that is real but not rational.

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".)

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".) � � � x 2 is rational � � � � x 3 is irrational � x ∈ R x ∈ R A = B = � � A ∩ B = { x ∈ R | ??? } A ∪ B = { x ∈ R | ??? }

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 }

Before next class... • Watch videos 1.4, 1.5, and 1.6 • Download the next class’s slides (no need to look at them!)