What defines a valid function?
A valid function is a set of pairs where each input x:
- Appears exactly once
- Has exactly one output y.
Valid function example: f = {(1,a), (2,b), (3,a)}.
Why is function invalid?
The set: {(1,a), (1,b)}
The set {(1,a), (1,b)} is invalid because the input 1 has two outputs.
This fails uniqueness.
This violates the definition of a function.
Totality requires every input in X to have an output.
How is the set {(1,a), (3,a)} invalid?
suppose the inputs are 1,2,3
The set {(1,a), (3,a)} is invalid because the input 2 has no output.
How is a function defined mathematically?
- A function f that maps elements of a set X to elements of a set Y is a subset of X×Y such that for every x∈X, there is exactly one y∈Y for which
(x,y)∈f.
What does f:X→Y signify?
It signifies that f is a function from set X to set Y.
What is the domain of a function?
The domain of a function is the set X from which inputs are taken.
What is the target of a function?
The target of a function is the set Y to which outputs are mapped.
What is an injective function?
A function f:X → Y is injective if there’s not a domain for every single y
Too little domains with an overabundant amount of outputs
What is a surjective function?
A function f:X → Y is surjective if for every y∈Y, there exists an x∈X such that f(x)=y. Too many domain is not capable of connecting to an output.
True or False: A function can have multiple outputs for a single input.
False.
A function must map each input to exactly one output.
How many values can 1 bit represent?
1 bit can represent 2 values (0 or 1).
Fill in the blank: Each bit can hold a value of either 0 or 1, meaning that with n bits, you can represent _______ different values.
2^n
What is a requirement for every block Ai in a universal set U?
Every block Ai has at least one element
This ensures that the blocks are nonempty.
What does it mean for blocks to be pairwise disjoint?
No element is in two different blocks
This prevents overlap between blocks.
What is the covering condition for a universal set U?
For every x ∈ U, there is exactly one Ai that contains x
This ensures that all elements of U are accounted for.
What is an equivalence relation in the context of partitioning a set?
A relation that groups elements into equivalence classes as blocks
This is a powerful method for constructing partitions.
What is a method to avoid endpoint overlap in intervals on ℝ?
Use half-open intervals like (-∞, a), {a}, (a, ∞)
This ensures every real number is included without overlap.
How can parity partition ℤ?
Parity (even/odd) partitions ℤ perfectly
This creates two distinct groups.
How are prime, composite, and 1 classified in partitioning?
1 is neither prime nor composite; give {1} its own block
Special attention is needed for the classification of 2.
What is a clean partition for ℝ?
(-∞, 0), {0}, (0, ∞)
This clearly separates negative, zero, and positive numbers.
How can strings be partitioned by length?
{strings of length n} for n ≥ 0; include the empty string as length 0 if U includes it
This categorizes strings based on their length.
What is a common way to partition product sets like ℝ²?
Quadrants plus axes: Q1, Q2, Q3, Q4, x-axis{0}, y-axis{0}, and {origin}
Each boundary must be its own block to prevent overlap.
What is a common pitfall regarding boundary overlaps?
Closed/closed at the same endpoint causes intersection
This can lead to elements being counted in multiple blocks.
