Functions

Set Identity: Study Notes

Simplified Definition

Set Identity is a mathematical rule showing that two different-looking set expressions are actually exactly the same. Think of it like "Algebra for Sets"—just as $2(x+3)$ is the same as $2x + 6$, these laws prove that certain combinations of sets always result in the same group of elements.

The Fundamental Laws of Set Identity

The "Big Three" Complex Laws

1. Distributive Law: Just like multiplying across parentheses.

   • $A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$

2. De Morgan’s Laws: How to distribute a complement (the "not" symbol).

   • $(A\cup B)^{c}=A^{c}\cap B^{c}(The"U"$ flip to → $"\cap")$

   • $(A\cap B)^{c}=A^{c}\cup B^{c}(The"\cap"$ flip to → $"U")$

3. Absorption Law: A set "absorbs" the other if it appears twice in a specific way.

   • $A \cup (A \cap B) = A$

Symmetric Differences: The symmetric difference between two sets AA and BB, denoted as AΔBAΔB, represents the elements that are in either of the sets but not in their intersection. In mathematical terms, it can be expressed as AΔB=(A∪B)∖(A∩B)AΔB=(A∪B)∖(A∩B). This operation is particularly useful in various applications where the unique elements of each set are of interest, providing insights into the differences between two groups.

Quick Guide: Injective, Surjective, and Bijective Functions

1. Core Definitions

• Injective (One-to-One): Every output $(y)$ comes from exactly one input $(x)$ . No two different inputs share the same output.

  • Visual Check: Passes the Horizontal Line Test.

• Surjective (Onto): The range equals the codomain. Every possible output in the target set is "hit" by at least one input.

  • Visual Check: The graph covers the entire vertical span of the codomain.

• Bijective: A function that is both Injective and Surjective. This means it is a perfect "one-to-one correspondence."

2. How to Prove or Disprove

To Prove Injective:

• The Algebraic Method: Assume $f(x_1) = f(x_2)$ . Solve the equation. If you can prove that $x_1$ must equal $x_2$, the function is injective.

• To Disprove: Find a "counterexample." Find two different $x$ values that give the same $y$ . (e.g., for $f(x) = x^2$, both $2$ and $-2$ result in $4$ ).

To Prove Surjective:

• The Range Method: Solve the function for $x$ in terms of y (e.g., $x = \sqrt{y-1}$ ). Check if this $x$ exists for every $y$ in the codomain.

• To Disprove: Find a $y$ value in the codomain that can never be reached. (e.g., for $f(x) = x^2$ where the codomain is all Real numbers, there is no $x$ that results in $-5$ ) .

Flashcards

Front: What is the definition of an Injective function?

Back: A "One-to-One" function where every element of the codomain is mapped to by at most one element of the domain. $(f(x_1) = f(x_2) \implies x_1 = x_2)$ .

Front: What does it mean for a function to be Surjective?

Back: It is "Onto," meaning every element in the codomain has at least one corresponding element in the domain. The Range = Codomain.

Front: What two conditions must be met for a function to be a Bijection?

Back: The function must be both Injective (one-to-one) and Surjective (onto).

Front: How do you disprove Injectivity?

Back: Find two distinct inputs $(x_1 \neq x_2)$ that produce the same output $(f(x_1) = f(x_2))$ .

Front: If a function's codomain is $\mathbb{R}$ (all real numbers) but its range is only positive numbers, is it Surjective?

Back: No. Because there are values in the codomain (negative numbers) that are never reached by the function.

Front: What is the Horizontal Line Test used for?

Back: To determine if a function is Injective. If any horizontal line crosses the graph more than once, it is NOT injective.

Front: What is the Identity Law for Union?

Back: $A \cup \emptyset = A$ . Adding nothing to a set leaves it unchanged.

Front: What are De Morgan’s Laws?

Back: 1) $(A \cup B)^c = A^c \cap B^c$ and $2) (A \cap B)^c = A^c \cup B^c$ . The complement of a union is the intersection of the complements (and vice versa).

Front: Define the Domination Law for Intersection.

Back: $A \cap \emptyset = \emptyset$ . Intersecting any set with an empty set results in an empty set.

Front: What is the Commutative Law?

Back: $A \cup B = B \cup A$ . The order in which you join or intersect sets does not change the result.

Front: What does $A \cup A^c = U$ represent?

Back: The Complementation Law. A set joined with everything not in that set equals the entire Universal set $(U)$.

Quick Tips for Context

• $\emptyset$ (Empty Set): Represents "zero" or nothing.

• $U$ (Universal Set): Represents "everything" in the context of the problem.

• $A^c$ or $A'$ (Complement): Everything that is not in $A$ .

• The "Flip" Trick: Whenever you apply a complement to a whole group (DeMorgan’s), always remember to flip the sign ($\cup$ becomes $\cap$ , and $\cap$ becomes $\cup$ ).