Section 2.2: Subsets

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Learning Objectives

• Define the complement of a set
• Find all subsets of a set
• Use subset notation

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Before we discuss anything further regarding sets, we need to define the universal set.

Definition: Universal Set, U

A Universal Set (denoted by U ) is the set that contains all elements under consideration in a particular mathematical discussion or problem.

• Key point: It's not "everything in existence"; it's "everything relevant to our current problem."

 

Every mathematical discussion about sets needs boundaries. Without defining our universal set first, we're like:

• A teacher talking about "students" without specifying which school
• A chef discussing "ingredients" without mentioning which kitchen
• A librarian organizing "books" without identifying which library

Imagine you're having a conversation about students at Orange Coast College. We would define the universal set:

U = {All students at OCC}

Everything you discuss (freshmen, math majors, athletes, etc.) comes from this universe. You wouldn't suddenly start talking about students from other colleges, because they are outside your "universe".

The universal set changes based on what we're discussing. Let's look at another situation.

In 2022, McDonald's teamed up with street-wear brand Cactus Plant Flea Market to create one of the most popular Happy Meal toy collections ever that was aimed at adults. The collection consisted of four figurines to collect: Grimace, Cactus Buddy, the Hamburglar, and Birdie. Here, we would define the universal set as:

U = {Grimace, Cactus Buddy, Grimace, Birdie}

 

Once we define our universal set U, several important relationships automatically become clear. We can now define those set relationships which are also referred to as set inclusions.

 Imagine you're organizing your music collection. You have:

• All your songs
• Just your jazz songs
• Just your workout songs
• Songs that are both jazz AND good for working out

These collections have natural relationships. Your jazz songs are part of your entire collection, and your jazz workout songs are part of your jazz collection. In mathematics, we call these relationships subsets.

Definition: Subset, ⊆

Set A is a subset of set B if every element in A is also in B.

Symbolically: A ⊆ B (read as "A is a subset of B ")

(Consequently, if set A is a not a subset of set B, then A ⊈ B.)

 

Example #2.2.1 🤔

Let A = {red, white} and B = {red, white, blue}. Decide whether the statement A ⊆ B is true or false.

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✅ Solution:

Since the elements red and white from set A are also contained in set B, then A is a subset of B.

Thus, the statement A ⊆ B is true.

Example #2.2.2 🤔

Let A = {red, white} and B = {red, white, blue}. Decide whether the statement B ⊆ A is true or false.

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✅ Solution:

Since there exists at least one element in set B (blue) that is NOT contained in set B, then B is not a subset of A.

Thus, the statement B ⊆ A is false. (Note: B ⊈ A is true).

Example #2.2.3 🤔

Let A = {Cities in the United States} and B = {Cities in California}. Decide whether the statement A ⊆ B is true or false.

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✅ Solution: 

Since there is at least one city in the United States (i.e., Chicago) that is NOT in California, then A is not a subset of B.

Thus, the statement A ⊆ B is false. (Note: A ⊈ B is true).

Example #2.2.4 🤔

Let A = {Cities in the United States} and B = {Cities in California}. Decide whether the statement B ⊆ A is true or false.

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✅ Solution:

Since every city in California is also a city in the United States, then B is a subset of A.

Thus, the statement B ⊆ A is true.

Example #2.2.5 🤔

Let A = {English majors at our college} and B = {All students at our college}. Decide whether the statement A ⊆ B is true or false.

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✅ Solution:

Since every English major is a student at the college, then A is a subset of B.

Thus, the statement A ⊆ B is true.

Example #2.2.6 🤔

Let A = {Legislative, Executive, Judicial} and B = {The three branches of U.S. government}. Decide whether the statement A ⊆ B is true or false.

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✅ Solution:

Since the Legislative, Executive, and Judicial branches are the three branches of U.S. government, then A is a subset of B

Thus, the statement A ⊆ B is true.

(Since set A and set  B are exactly the same elements, A = B is also true.)

Example #2.2.7 🤔

Let A = {1, 2} and B = {1, 2, 3}. Decide whether the statement A ⊆ B is true or false.

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✅ Solution:

Since 1 is an element of B and 2 is and element of B, (or equivalently 1 ∈ B and 2 ∈ B ), then A is a subset of B.

Thus, the statement A ⊆ B is true.

Example #2.2.8 🤔

Let A = {1, 2} and B = {1, 2, 3}. Decide whether the statement B ⊆ A is true or false.

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✅ Solution:

Since 3 is NOT an element of B, (or equivalently 3 ∉ B ), then B is not a subset of A.

Thus, the statement B ⊆ A is false. (Note: B ⊈ A is true).

Section 2.2: Subsets [In-Class Exercises]

1. Let A = {1, 2, 3, 4} and B = {1, 2, 3, 4, 5}. Decide whether the statement A ⊆ B is true or false.
2. Let C = {circle, triangle, rhombus} and D = {circle, triangle, hexagon}. Decide whether the statement C ⊆ D is true or false.
3. Let E = {a, b, c, x, y} and F = {a, b, c}. Decide whether the statement E ⊈ F is true or false.
4. In Major League Baseball (MLB), there are 30 teams among 6 divisions. The American League West division contains the following five teams: Angels, Astros, Athletics, Mariners and Rangers. Let M = {All 30 MLB Baseball teams} and W = {Five American League West teams}. Which statement is true, M ⊆ W or W ⊆ M ?
5. Let S = {All Star Wars movies/films} and F = {All science fiction/fantasy movies/films}. Which statement is true, F ⊆ S or S ⊆ F ?

Answers

1) True;   2) False;   3) False;   4) W ⊆ M;   5) S ⊆ F.

 

Now, let's look at another mathematical example, where both C and D are the exact same sets. Let C = {1, 2, 3} and D = {1, 2, 3}. Is C ⊆ D ? Let's check.

• Is 1 ∈ D? Yes ✓
• Is 2 ∈ D? Yes ✓
• Is 3 ∈ D? Yes ✓

Thus, C ⊆ D is true! This feels different from our first mathematical example above.

 

So, let's compare the two mathematical examples

• Earlier: A = {1, 2} and B = {1, 2, 3}
• Here A ⊆ B, but A is clearly "smaller" than B (as it relates to cardinality)
• Now: C = {1, 2, 3} and D = {1, 2, 3}
• Here C ⊆ D, but C and D are identical!

This observation led mathematicians to create a more specific term. When we want to say that A is a subset of B, but A is definitely not equal to B, meaning that B has at least one element that A doesn't have, then we call this a proper subset.

 

Definition: Proper Subset, ⊂

A is a proper subset of B (written A ⊂ B) when:

1. A ⊆ B (A is a subset of B ), AND
2. A ≠ B (A is not equal to B )

Symbolically: A ⊂ B (read as "A is a proper subset of B ")

(Consequently, if set A is a not a proper subset of set B, then A ⊄ B.)

 

So, in other words, for A ⊂ B, means that there is at least one element in set B (the set on the right) that is not in set A (the set on the left. The key difference is that a subset (⊆) allows for the possibility that A = B, whereas a proper subset (⊂), has A "strictly smaller" than B.

 

Example #2.2.9 🤔

Let A = {red, white} and B = {red, white, blue}. Decide whether the statement A ⊂ B is true or false.

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✅ Solution:

Check the following conditions:

1. Is A ⊆ B ? Yes, every element of set A is also contained in set B. ✅
2. Is A ≠ B ? Yes, set A is not equivalent to set B. ✅

Since both conditions are satisfied, then A is a proper subset of B.

Thus, the statement A ⊂ B is true.

Example #2.2.10 🤔

Let A = {red, white} and B = {red, white, blue}. Decide whether the statement B ⊂ A is true or false.

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✅ Solution:

Check the following conditions:

1. Is B ⊆ A ? No, every element of set B is NOT contained in set A. ❌
2. Is A ≠ B ? Yes, set A is not equivalent to set B. ✅ (Note: There is no need to check for the second condition, if the first one fails.)

Since at least one of the conditions are NOT satisfied, then B is not a proper subset of A.

Thus, the statement B ⊂ A is false. (Note: B ⊄ A is true).

Example #2.2.11 🤔

Let A = {Cities in the United States} and B = {Cities in California}. Decide whether the statement A ⊂ B is true or false.

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✅ Solution:

Check the following conditions:

1. Is A ⊆ B ? No, every element of set A is NOT contained in set B. ❌
2. Is A ≠ B ? Yes, set A is not equivalent to set B. ✅ (Note: There is no need to check for the second condition, if the first one fails.)

Since at least one of the conditions are NOT satisfied, then A is not a proper subset of B.

Thus, the statement A ⊂ B is false. (Note: A ⊄ B is true).

Example #2.2.12 🤔

Let A = {Cities in the United States} and B = {Cities in California}. Decide whether the statement B ⊂ A is true or false.

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✅ Solution:

Check the following conditions:

1. Is B ⊆ A ? Yes, every element of set B is also contained in set A. ✅
2. Is A ≠ B ? Yes, set A is not equivalent to set B. ✅

Since both conditions are satisfied, then B is a proper subset of A.

Thus, the statement B ⊂ A is true.

Example #2.2.13 🤔

Let A = {English majors at our college} and B = {All students at our college}. Decide whether the statement A ⊂ B is true or false.

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✅ Solution:

Check the following conditions:

1. Is A ⊆ B ? Yes, every element of set A is also contained in set B. ✅
2. Is A ≠ B ? Yes, set A is not equivalent to set B. ✅

Since both conditions are satisfied, then A is a proper subset of B.

Thus, the statement A ⊂ B is true.

Example #2.2.14 🤔

Let A = {Legislative, Executive, Judicial} and B = {The three branches of U.S. government}. Decide whether the statement A ⊂ B is true or false.

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✅ Solution:

Check the following conditions:

1. Is A ⊆ B ? Yes, every element of set A is also contained in set B. ✅
2. Is A ≠ B ? No, set A is equivalent to set B. ❌

Since at least one of the conditions are NOT satisfied, then A is not a proper subset of B.

Thus, the statement A ⊂ B is false. (Note: A ⊄ B is true).

(Think of a proper subset as if is was a strictly less than, but not equal to, sign (<). So, is the set {1, 2, 3} < {1, 2, 3}? No.)

Example #2.2.15 🤔

Let A = {1, 2} and B = {1, 2, 3}. Decide whether the statement A ⊂ B is true or false.

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✅ Solution:

Check the following conditions:

1. Is A ⊆ B ? Yes, every element of set A is also contained in set B. ✅
2. Is A ≠ B ? Yes, set A is not equivalent to set B. ✅

Since both conditions are satisfied, then A is a proper subset of B.

Thus, the statement A ⊂ B is true.

(Think of a proper subset as if is was a strictly less than, but not equal to, sign (<). So, is the set {1, 2} < {1, 2, 3}? Yes.)

Example #2.2.16 🤔

Let A = {1, 2} and B = {1, 2, 3}. Decide whether the statement B ⊂ A is true or false.

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✅ Solution:

Check the following conditions:

1. Is B ⊆ A ? No, every element of set B is NOT contained in set A. ❌
2. Is A ≠ B ? Yes, set A is not equivalent to set B. ✅ (Note: There is no need to check for the second condition, if the first one fails.)

Since at least one of the conditions are NOT satisfied, then A is not a proper subset of B.

Thus, the statement B ⊂ A is false. (Note: B ⊄ A is true).

(Think of a proper subset as if is was a strictly less than, but not equal to, sign (<). So, is the set {1, 2, 3} < {1, 2}? No.)

Example #2.2.17 🤔

Let A = {x, y, z} and B = {x, y, z}. Decide whether the statement A ⊂ B is true or false.

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✅ Solution

Check the following conditions:

1. Is A ⊆ B ? Yes, every element of set A is also contained in set B. ✅
2. Is A ≠ B ? No, set A is equivalent to set B. ❌

Since at least one of the conditions are NOT satisfied, then A is not a proper subset of B.

Thus, the statement A ⊂ B is false. (Note: A ⊄ B is true).

(Think of a proper subset as if is was a strictly less than, but not equal to, sign (<). So, is the set {x, y, z} < {x, y, z}? No.)

Section 2.2: Subsets [In-Class Exercises]

6. Let A = {1, 2, 3, 4} and B = {1, 2, 3, 4, 5}. Decide whether the statement A ⊂ B is true or false.
7. Let C = {circle, triangle, hexagon} and D = {circle, triangle, octagon}. Decide whether the statement D ⊂ C is true or false.
8. Let E = {a, b, c} and F = {a, b, c, x, y}. Decide whether the statement E ⊄ F is true or false.
9. Let G = {2, 4, 6, 8, 10} and H = {2, 4, 6, 8, 10}. Decide whether the statement G ⊂ H is true or false.
10. In Major League Baseball (MLB), there are 30 teams among 6 divisions. The National League West division contains the following five teams: Diamondbacks, Dodgers, Giants, Padres and Rockies. Let M = {All 30 MLB Baseball teams} and W = {Five National League West teams}. Which statement is true, M ⊂ W or W ⊂ M or neither?
11. Let S = {All planets in our solar system} and P = {Mercury, Venus, Earth}. Which statement is true, P ⊂ S or S ⊂ P or neither.

Answers

6) True;   7) False;   8) False;   9) False;   10) W ⊂ M;   11) P ⊂ S.

 

Properties of Subsets and Proper Subsets

1. Every Set is a Subset of Itself

• {1, 2, 3} ⊆ {1, 2, 3} ✓
• But {1, 2, 3} ⊄ {1, 2, 3} (not a proper subset of itself)

 
2. The Empty Set is a Subset of Every Set

• ∅ ⊆ {1, 2, 3} ✓
• ∅ ⊆ {cats, dogs} ✓
• ∅ ⊆ ∅ ✓

Why? Because there are no elements in ∅ to violate the subset condition!

3. Transitivity

If A ⊆ B and B ⊆ C, then A ⊆ C

• A = {Calculus students}
• B = {Math majors}
• C = {All students}
• A ⊆ B ⊆ C, so A ⊆ C

 

Common Mistakes to Avoid

Mistake 1: Confusing ∈ and ⊆

• ❌ Incorrect: {2} ∈ {1, 2, 3}
• ✅ Correct: {2} ⊆ {1, 2, 3} or 2 ∈ {1, 2, 3}

Mistake 2: Forgetting the Empty Set

• When listing all subsets of {1, 2}, don't forget: ∅, {1}, {2}, {1, 2}

Mistake 3: Thinking Proper Subsets Can Be Equal 

• If A ⊂ B, then A ≠ B (they cannot be equal)   

 

Conclusion:

Understanding subsets helps us:

• Organize information logically
• Understand hierarchical relationships
• Make precise statements about collections of objects
• Build foundation for more advanced mathematical concepts

The distinction between subsets (⊆) and proper subsets (⊂) might seem subtle, but it's crucial for mathematical precision - just like the difference between "less than or equal to" (≤) and "less than" (<).

Remember: Every proper subset is a subset, but not every subset is a proper subset!

 

Example #2.2.18 🤔

Let A = {1, 2, 3, 4}, B = {1, 2, 3, 4, 5}, C = {1, 2, 3, 4, 5}, and D = {1, 2, 3, 4, 5, ….}. Decide whether each of the statements is true or false.

1. A ⊆ B
2. B ⊆ A
3. A ⊂ B
4. B ⊂ A
5. B ⊆ C
6. C ⊆ B
7. B ⊂ C
8. C ⊂ B
9. B ⊆ D
10. D ⊆ B
11. B ⊂ D
12. D ⊂ B

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✅ Solution:

1. True
2. False
3. True
4. False
5. True
6. True
7. False
8. False
9. True
10. False
11. True
12. False

Example #2.2.19 🤔

Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, A = {1, 2}, B = {3, 4}, and C = {1, 2, 3, 4}. Decide whether each of the statements is true or false.

1. A ⊆ B
2. B ⊆ A
3. A ⊂ B
4. B ⊂ A
5. A ⊆ C
6. C ⊆ A
7. A ⊂ C
8. C ⊂ A
9. A ⊆ U
10. U ⊆ A
11. U ⊂ A
12. A ⊂ U

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✅ Solution:

1. False
2. False
3. False
4. False
5. True
6. False
7. True
8. False
9. True
10. False
11. False
12. True

Section 2.2: Subsets [In-Class Exercises]

12. Let U = {1, 2, 3, 4, 5, 6}, A = {3, 4, 5}, B = {1, 3, 6}, C = {3, 4, 5, 6}, and D = {1, 3, 6}. Decide whether each of the statements is true or false.

1. A ⊆ C
2. C ⊆ A
3. A ⊂ C
4. C ⊂ A
5. B ⊆ D
6. D ⊆ B
7. B ⊂ D
8. D ⊂ B
9. A ⊆ B
10. B ⊂ A
11. A ⊂ U
12. U ⊂ A

Answers

a) True;   b) False;   c) True;   d) False;   e) True;   f) True;   g) False;   h) False;   i) False;   j) False;   k) True;   l) False.

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