This document introduces set theory concepts like inclusion, union, intersection and difference. It presents three facts about the monotonicity of set operations, and proves the first fact - that if sets X1 and Y1 are subsets of sets X2 and Y2 respectively, then the union of X1 and Y1 is a subset of the union of X2 and Y2. The document also provides exercises to prove similar results and identify true or false statements involving set relationships.

1. Introduction to set theory and to methodology and philosophy of
mathematics and computer programming
Properties of inclusion
An overview
by Jan Plaza
c 2017 Jan Plaza
Use under the Creative Commons Attribution 4.0 International License
Version of February 14, 2017

2. Fact (on monotonicity of set operations)
1. If X1 ⊆ X2 and Y1 ⊆ Y2 then X1 ∪ Y1 ⊆ X2 ∪ Y2.
2. If X1 ⊆ X2 and Y1 ⊆ Y2 then X1 ∩ Y1 ⊆ X2 ∩ Y2.
3. If X1 ⊆ X2 and Y1 ⊇ Y2 then X1 − Y1 ⊆ X2 − Y2.

3. Fact (on monotonicity of set operations)
1. If X1 ⊆ X2 and Y1 ⊆ Y2 then X1 ∪ Y1 ⊆ X2 ∪ Y2.
2. If X1 ⊆ X2 and Y1 ⊆ Y2 then X1 ∩ Y1 ⊆ X2 ∩ Y2.
3. If X1 ⊆ X2 and Y1 ⊇ Y2 then X1 − Y1 ⊆ X2 − Y2.
Recall:
X ⊆ Y iff ∀u (u ∈ X → u ∈ Y ).
u ∈ X ∪ Y iff u ∈ X or u ∈ Y .

4. 1. If X1 ⊆ X2 and Y1 ⊆ Y2 then X1 ∪ Y1 ⊆ X2 ∪ Y2.
Proof

5. 1. If X1 ⊆ X2 and Y1 ⊆ Y2 then X1 ∪ Y1 ⊆ X2 ∪ Y2.
Proof
Assume X1 ⊆ X2 and Y1 ⊆ Y2.
Goal: X1 ∪ Y1 ⊆ X2 ∪ Y2.

6. 1. If X1 ⊆ X2 and Y1 ⊆ Y2 then X1 ∪ Y1 ⊆ X2 ∪ Y2.
Proof
Assume X1 ⊆ X2 and Y1 ⊆ Y2.
Goal: X1 ∪ Y1 ⊆ X2 ∪ Y2.
Take any u.
Assume u ∈ X1 ∪ Y1.
New goal: u ∈ X2 ∪ Y2.

7. 1. If X1 ⊆ X2 and Y1 ⊆ Y2 then X1 ∪ Y1 ⊆ X2 ∪ Y2.
Proof
Assume X1 ⊆ X2 and Y1 ⊆ Y2.
Goal: X1 ∪ Y1 ⊆ X2 ∪ Y2.
Take any u.
Assume u ∈ X1 ∪ Y1.
New goal: u ∈ X2 ∪ Y2.
Case 1: u ∈ X1.
Case 2: u ∈ Y1.

8. 1. If X1 ⊆ X2 and Y1 ⊆ Y2 then X1 ∪ Y1 ⊆ X2 ∪ Y2.
Proof
Assume X1 ⊆ X2 and Y1 ⊆ Y2. (*)
Goal: X1 ∪ Y1 ⊆ X2 ∪ Y2.
Take any u.
Assume u ∈ X1 ∪ Y1.
New goal: u ∈ X2 ∪ Y2.
Case 1: u ∈ X1.
u ∈ X2, by (*).
Case 2: u ∈ Y1.

9. 1. If X1 ⊆ X2 and Y1 ⊆ Y2 then X1 ∪ Y1 ⊆ X2 ∪ Y2.
Proof
Assume X1 ⊆ X2 and Y1 ⊆ Y2. (*)
Goal: X1 ∪ Y1 ⊆ X2 ∪ Y2.
Take any u.
Assume u ∈ X1 ∪ Y1.
New goal: u ∈ X2 ∪ Y2.
Case 1: u ∈ X1.
u ∈ X2, by (*).
So, u ∈ X2 ∪ Y2, and we reached the goal.
Case 2: u ∈ Y1.

10. 1. If X1 ⊆ X2 and Y1 ⊆ Y2 then X1 ∪ Y1 ⊆ X2 ∪ Y2.
Proof
Assume X1 ⊆ X2 and Y1 ⊆ Y2. (*)
Goal: X1 ∪ Y1 ⊆ X2 ∪ Y2.
Take any u.
Assume u ∈ X1 ∪ Y1.
New goal: u ∈ X2 ∪ Y2.
Case 1: u ∈ X1.
u ∈ X2, by (*).
So, u ∈ X2 ∪ Y2, and we reached the goal.
Case 2: u ∈ Y1.
Similar.

11. Exercise
1. Repeat the proof above on your own. Write case 2 in full.
2. Prove: If X1 ⊆ X2 and Y1 ⊆ Y2 then X1 ∩ Y1 ⊆ X2 ∩ Y2.
3. Disprove: If X1 ⊂ X2 and Y1 ⊂ Y2 then X1 ∪ Y1 ⊂ X2 ∪ Y2.

12. Exercise
True or false?
1. X ⊆ Y iff X ∪ Y =Y .
2. X ⊇ Y iff X ∩ Y =Y .
3. X ⊆ X ∪ Y and Y ⊆ X ∪ Y .
4. X ⊇ X ∩ Y and Y ⊇ X ∩ Y .
5. X ⊆ Z and Y ⊆ Z iff X ∪ Y ⊆ Z.
6. X ⊇ Z and Y ⊇ Z iff X ∩ Y ⊇ Z.