Selected items from the Introduction to set theory and to methodology and philosophy of mathematics and computer programming.

1. Introduction to set theory and to methodology and philosophy of
mathematics and computer programming
Symmetric difference
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of February 6, 2017

2. Definition
The symmetric difference of X and Y , denoted X Y , is the set
(X − Y ) ∪ (Y − X).
(Think of it as {u : u ∈ X ⊕ u ∈ Y }, where ⊕ is the exclusive-or.)
X Y
Example
{1, 2} {2, 3} = {1, 3}.

3. Fact
X Y = (X ∪ Y ) − (X ∩ Y ).
Fact
1. z ∈ X Y iff z ∈ X ∪ Y and z ∈ X ∩ Y .
2. z ∈ X Y iff z ∈ X ∪ Y or z ∈ X ∩ Y .

4. Implementation
1. As X Y = (X − Y ) ∪ (Y − X),
with sets implemented as dynamic hash tables,
symmetric difference can be defined in terms of union and set-difference.
2. As z ∈ X Y iff z ∈ X ⊕ z ∈ Y ,
with bit-array implementation of sets,
one can use the bitwise exclusive-or (xor) operation, often denoted by ^.

5. Exercise
1. X ∅=...
2. X U =...
3. X X =...
4. X Xc =...
Exercise
Complete:
1. (X ∪ Y ) − (X ∩ Y )=...
2. (X ∩ Y ) ∪ (X Y )=...
3. (X − Y ) ∪ (Y − X)=...

6. Exercise
1. X ⊆ Y iff X Y =...
2. X ⊆ Y iff X Y c =...
3. X ⊆ Y iff Xc Y =...
4. X ⊆ Y iff Xc Y c =...