- “Choose an element” method for proving something about all members of a
set requires just one element?
- Yes, but that element has to be generic in the sense that you don’t assume any properties that all elements of the set don’t have
- For instance, if set is defined as being blue things you can assume the element
is blue, but nothing about its shape, size, etc.
- Notation: AC = U - A
- Notation for the odd natural numbers?
- Math, natural language, and (mostly) LaTeX versions
- Math, natural language, and (mostly) LaTeX versions
- Prove that { x ∈ Z | x ≡ 3 (mod 12) } is disjoint from
{ y ∈ Z | y ≡ 2 (mod 8) } (progress check 5.15)
- Assume { x ∈ Z | x ≡ 3 (mod 12) } ∩ { y ∈ Z | y ≡ 2 (mod 8) } ≠ ∅
- So there exists integer a such that a-3 = 12k and a-2 = 8c for integers c, k
- (a = 12k + 3 and a = 8c + 2 from division algorithm)
- Show k and c can’t both be integers…
- a-2 = 12k + 1 = 8c
- c = 3/2 k + 1/8
- Case 1: k = 2b for integer b
- 3/2 k + 1/8 = 3b + 1/8 so c not integer!
- Case 2: k = 2b + 1
- 3/2 k + 1/8 = 3b + 1/2 + 1/8, c not integer!
- Or a shorter proof
- 12k + 3 = 8c + 2
- 12k + 1 = 8c
- 2(6k) + 1 = 2(4c)
- Odd = even, contradiction!
- The algebra of sets
- Read section 5.3