- REU (= “Research Experiences for Undergraduates”) talk
- Thursday, Nov. 10, 3:30, Newton 213
- Gabby Angeloro on the research she did in an REU this past summer
- Extra credit for going and writing paragraph re how you relate to it
- Ordered triples, etc.
- Formally (a,b,c) = ( (a,b), c )
- So A × B × C × … = (...(A×B) × C) × … )
- Finish proof that (A - B) × C = (A × C) - (B × C)
- Last time we proved that (A - B) × C ⊆ (A × C) - (B × C)
- This time… (A × C) - (B × C) ⊆ (A - B) × C
- (A × C) - (B × C) = (A × C) ∩ (B × C)C
- Let u = (x,y) ∈ (A × C) - (B × C)
- So x ∈ A and y ∈ C and (x,y) ∉ (B × C)
- So x ∈ A and y ∈ C and ( x ∉ B or y ∉ C)
- So x ∈ A and y ∈ C and x ∉ B
- So x ∈ A-B and y ∈ C
- So (x,y) = u ∈ (A-B) × C
- Section 5.5
- Let U be the set of students in this course, and consider the subsets of U that
can be made by grouping students according to major
- Is the result a family of sets?
- “Family of sets” = set of sets
- So yes
- For example, using names for the sets: { Math, Acct, Bio, Phys }
- = { {SY,MQ,DH,SN,TC,FC,VL,JY,CP,TS}, {BP}, {ST}, … } as a set of sets
- Indexed family = family whose members are identified by labels aka
“indices”
- We could write the above as { A1, A2, AFOO, A3.14 }
- With A1 = Math, A2 = Acct, AFOO = Bio, A3.14 = Phys
- Index set = Λ = { 1, 2, FOO, 3.14 }
- Is the result a family of sets?
- Let 𝒜 = { An | n ∈ ℕ } and define Ai as
{ -i, -(i-1), …, i-1, i }
- What is ∪i=13 Ai?
- How about ∪i=1∞ Ai?
- How about ∩i=1∞ Ai?
- Consider 𝒜 = { Ax | x ∈ ℝ } where Ax = {x2}
- Is this an indexed family? Yes, even though each member only has 1 element
- Intersection of all members of this family is the empty set, i.e., it’s just coincidence that the intersection of all members above was the first member
- Introduction to functions
- Read section 6.1