Set Theory: Operations, Complements, and De Morgan's Laws
Commutativity of Set Operations
Intersection: Order does not matter for intersection. A \cap B = B \cap A.
Union: Order does not matter for union. A \cup B = B \cup A.
Set Difference
Definition: A - B represents all elements that belong to set A and do not belong to set B. Symbolically, it's expressed as x \in A \text{ and } x \notin B.
Alternative Representation: The set difference A - B can be rewritten as the intersection of A and the complement of B (B^c): A - B = A \cap B^c.
The complement of B (B^c) includes all elements that are not in B.
Non-Commutativity: Set difference is not commutative; A - B \neq B - A.
Example: If A = {4, 5, 6} and B = {2, 4, 6}, then A - B = {5} and B - A = {2}. These are clearly not equal.
Definition of Even and Odd Numbers
Even Numbers: A number is even if it is divisible by 2 with no remainder.
Odd Numbers: A number is odd if, when divided by 2, it yields a remainder of 1.
Classification of Zero: Zero (0) is an even number because 0 \div 2 = 0 with no remainder. (2 \times 0 = 0).
Set Complements and Examples (Universe: U = {0, 1, 2, 3, 4, 5, 6})
Complement of the Universe: The complement of the universe (U^c) is the empty set (\emptyset), as it contains no elements. (U^c = \emptyset).
Even and Odd Sets: Let E = {0, 2, 4, 6} (even numbers) and O = {1, 3, 5} (odd numbers).
The complement of the set of even numbers is the set of odd numbers (E^c = O).
The complement of the set of odd numbers is the set of even numbers (O^c = E).
Example Sets: Let A = {4, 5, 6} and B = {2, 4, 6}.
A^c = {0, 1, 2, 3} (elements in U but not in A).
B^c = {0, 1, 3, 5} (elements in U but not in B).
Set Operations with Examples
Intersection of Even and Odd Sets: The intersection of even and odd numbers is the empty set (E \cap O = \emptyset), as they share no common elements.
Disjoint Sets: If the intersection of two sets is empty, the sets are said to be disjoint.
Union of Even and Odd Sets: The union of even and odd numbers comprises all numbers in the universe (E \cup O = U).
Intersection of A and B: A \cap B = {4, 6}. These are the elements common to both A and B.
Complement of Intersection: (A \cap B)^c represents all elements in the universe that are not in the intersection of A and B. So, (A \cap B)^c = {0, 1, 2, 3, 5}.
Union of A and B: A \cup B = {2, 4, 5, 6}. This includes all unique elements from A or B.
Complement of Union: (A \cup B)^c represents all elements in the universe that are not in the union of A and B. So, (A \cup B)^c = {0, 1, 3}.
Subsets and Proper Subsets
Subset (B \subseteq E): Set B is a subset of set E if every element in B is also an element in E.
Proper Subset (B \subset E): Set B is a proper subset of set E if every element in B is also in E, AND there is at least one element in E that is not in B.
Example: Let B = {2, 4, 6} and E = {0, 2, 4, 6}. B is a proper subset of E because every element in B is in E, and 0 is in E but not in B.
Properties related to proper subsets:
If B \subset E, then B \cap E = B.
If B \subset E, then B \cup E = E.
De Morgan's Laws for Set Theory
These laws relate unions, intersections, and complements.
First Law: The complement of the intersection of two sets is equal to the union of their complements: (A \cap B)^c = A^c \cup B^c.
Verification (using previous examples):
(A \cap B)^c = {0, 1, 2, 3, 5}.
A^c \cup B^c = {0, 1, 2, 3} \cup {0, 1, 3, 5} = {0, 1, 2, 3, 5}. (The results match).
Second Law: The complement of the union of two sets is equal to the intersection of their complements: (A \cup B)^c = A^c \cap B^c.
Verification (using previous examples):
(A \cup B)^c = {0, 1, 3}.
A^c \cap B^c = {0, 1, 2, 3} \cap {0, 1, 3, 5} = {0, 1, 3}. (The results match).
Venn Diagrams with Two Sets (A and B)
A Venn diagram with two sets, A and B, divides the universal rectangle into 4 distinct regions.
Region 1: Elements in A but not in B (A \cap B^c).
Region 2: Elements in both A and B (A \cap B).
Region 3: Elements in B but not in A (A^c \cap B).
Region 4: Elements neither in A nor in B (A^c \cap B^c or (A \cup B)^c).
Region Representations:
A is represented by Regions 1 and 2.
A^c (Not A) is represented by Regions 3 and 4.
B is represented by Regions 2 and 3.
B^c (Not B) is represented by Regions 1 and 4.
A \cap B is represented by Region 2.
(A \cap B)^c is represented by Regions 1, 3, 4.
A \cup B is represented by Regions 1, 2, 3.
(A \cup B)^c is represented by Region 4.
Venn Diagram Proof of De Morgan's Laws:
(A \cap B)^c (Regions 1, 3, 4) is equivalent to A^c \cup B^c ({3, 4} \cup {1, 4} = {1, 3, 4}).
(A \cup B)^c (Region 4) is equivalent to A^c \cap B^c (\text{not } A = {3, 4}, ext{not } B = {1, 4}, so {3, 4} \cap {1, 4} = {4}).
Commutativity of Set Operations
Intersection: When you intersect sets, the order doesn't change the result. For example, if you find what's common in A and then B, it's the same as finding what's common in B and then A. A \cap B = B \cap A.
Union: When you combine sets, the order doesn't change the result. Combining A with B gives the same result as combining B with A. A \cup B = B \cup A.
Set Difference
What it means: A - B means all the things that are in set A but are not in set B. We write this as "x is in A AND x is NOT in B" (x \in A \text{ and } x \notin B).
Another way to write it: You can also think of A - B as everything in A that is not in B. This is the same as the intersection of A with the complement of B (B^c). So, A - B = A \cap B^c. The complement of B (B^c) means everything outside of B in your entire group of possible elements.
Order matters: Set difference is not commutative. This means A - B is usually not the same as B - A.
Example: If A = {4, 5, 6} and B = {2, 4, 6}, then A - B = {5} (only 5 is in A but not B). And B - A = {2} (only 2 is in B but not A). As you can see, {5} \neq {2}.
What are Even and Odd Numbers?
Even Numbers: A number is even if you can divide it by 2 and have nothing left over.
Odd Numbers: A number is odd if, when you divide it by 2, there's always 1 left over.
Is Zero Even or Odd?: Zero (0) is an even number because 0 \div 2 = 0 with no remainder. (Think of it as 2 \times 0 = 0).
Set Complements (using Universe: U = {0, 1, 2, 3, 4, 5, 6})
Complement of the Universe: Everything not in the universe. Since the universe already contains everything, its complement is nothing (the empty set), written as U^c = \emptyset.
Even and Odd Sets: Let E = {0, 2, 4, 6} (our even numbers) and O = {1, 3, 5} (our odd numbers).
The complement of even numbers (E^c) is the set of odd numbers (O).
The complement of odd numbers (O^c) is the set of even numbers (E).
More Examples: Let A = {4, 5, 6} and B = {2, 4, 6}.
A^c = {0, 1, 2, 3} (these are the numbers in U that are not in A).
B^c = {0, 1, 3, 5} (these are the numbers in U that are not in B).
Understanding Set Operations with Examples
Even and Odd Intersection: What numbers are both even AND odd? None! So, their intersection is the empty set (E \cap O = \emptyset).
Disjoint Sets: If two sets have absolutely nothing in common (their intersection is empty), we call them
disjoint sets.Even and Odd Union: If you combine all even numbers and all odd numbers in our universe, you get the entire universe of numbers (E \cup O = U).
Intersection of A and B: The elements common to both A and B are A \cap B = {4, 6}.
Complement of Intersection: (A \cap B)^c means everything in the universe except what's common to A and B. So, (A \cap B)^c = {0, 1, 2, 3, 5}.
Union of A and B: A \cup B means all the unique elements found in either A or B (or both). So, A \cup B = {2, 4, 5, 6}.
Complement of Union: (A \cup B)^c means everything in the universe except what's in A or B. So, (A \cup B)^c = {0, 1, 3}.
What are Subsets?
Subset (B \subseteq E): Set B is a subset of set E if every single element in B can also be found in E.
Proper Subset (B \subset E): Set B is a proper subset of set E if every element in B is also in E, AND set E has at least one extra element that is not in B (meaning B is strictly smaller than E).
Example: Let B = {2, 4, 6} and E = {0, 2, 4, 6}. B is a proper subset of E because all elements of B are in E, but E also has 0, which isn't in B.
Subset Properties:
If B is a proper subset of E, then what's common between B and E is just B itself (B \cap E = B).
If B is a proper subset of E, then combining B and E gives you E itself (B \cup E = E).
De Morgan's Laws (Set Rules)
These rules show how complements, unions, and intersections are related.
First Law: If you take the complement of the common part of two sets (A and B), it's the same as combining what's not in A with what's not in B: (A \cap B)^c = A^c \cup B^c.
Check (with our examples):
(A \cap B)^c = {0, 1, 2, 3, 5}.
A^c \cup B^c = {0, 1, 2, 3} \cup {0, 1, 3, 5} = {0, 1, 2, 3, 5}. (They match!)
Second Law: If you take the complement of the combined part of two sets (A and B), it's the same as finding what's common between what's not in A and what's not in B: (A \cup B)^c = A^c \cap B^c.
Check (with our examples):
(A \cup B)^c = {0, 1, 3}.
A^c \cap B^c = {0, 1, 2, 3} \cap {0, 1, 3, 5} = {0, 1, 3}. (They match!)
Venn Diagrams (2 Circles - A and B)
A Venn diagram with two overlapping circles (A and B) divides the main box (the universe) into 4 distinct areas.
Area 1: Things only in A (not in B) - (A \cap B^c).
Area 2: Things in both A and B (the overlap) - (A \cap B).
Area 3: Things only in B (not in A) - (A^c \cap B).
Area 4: Things neither in A nor in B (outside both circles) - (A^c \cap B^c or (A \cup B)^c).
How Areas Show Set Operations:
Set A is Areas 1 and 2.
Things not in A (A^c) are Areas 3 and 4.
Set B is Areas 2 and 3.
Things not in B (B^c) are Areas 1 and 4.
The common part of A and B (A \cap B) is Area 2.
Things not common to A and B ((A \cap B)^c) are Areas 1, 3, 4.
The combined part of A and B (A \cup B) is Areas 1, 2, 3.
Things not in A or B ((A \cup B)^c) is Area 4.
Venn Diagram Proof of De Morgan's Laws (simple version):
(A \cap B)^c (Areas 1, 3, 4) is the same as combining what's not in A ({3, 4}) with what's not in B ({1, 4}), which gives us {1, 3, 4}. (Matches First Law)
$$(A \cup