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
Disjoint, covering and complementary sets
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of February 10, 2017

2. Disjoint sets
Definition
X and Y are disjoint if X ∩ Y = ∅.
Visualize disjoint sets:
X Y

3. Covering sets
Definition
X, Y cover U if X ∪ Y = U.
If U is known form the context, we just say X, Y are covering sets .
Visualize covering sets:
Y
X

4. Complementary sets
Definition
X, Y are complementary subsets of U if X, Y are disjoint and X, Y cover U.
If U is known form the context, we just say X, Y are complementary .
Visualize complementary sets:
X
Y

5. Exercises
True or false?
1. X, Y are complementary subsets of U iff Y = Xc.
2. X, Y cover U iff Xc, Y c are disjoint.
3. X, Y are complementary subsets of U iff Xc, Y c are complementary subsets of U.
4. ∅ and X are disjoint.
5. X and Xc are disjoint.
6. X − Y and Y are disjoint.
7. X − Y and X ∩ Y are disjoint.

6. Exercises
1. Disprove: if X ∩ Y = ∅ then X Y .
2. Complete: if X ∩ Y = ∅ and ... then X Y .
3. Disprove: if X ∪ Y = U then X Y .
4. Complete: if X ∪ Y = U and ... then X Y .