Kindle Notes & Highlights
0.2 Cartesian product One from column A, one from column B. The Cartesian product of two sets A and B is the set of all pairs (a, b) where a ∈ A and b ∈ B.
Proposition 0.2.3: For finite sets A and B, |A × B| = |A|·|B|. Quiz 0.2.4: What is the cardinality of {1, 2, 3,..., 10, J, Q, K} × {♥, ♠, ♣, ♦}?
The set D of possible inputs is called the domain of the function.
Example 0.3.3: Caesar was said to have used a cryptosystem in which each letter was replaced with the one three steps forward in the alphabet (wrapping around for X,Y, and Z).a Thus the plaintext MATRIX would be encrypted as the cyphertext PDWULA. The function that maps each plaintext letter to its cyphertext replacement could be written as A )→ D, B )→ E,C )→ F, D )→ G, W )→ Z, X )→ A, Y )→ B,Z )→ C This function’s domain and co-domain are both the alphabet {A, B, . . . , Z}.
composite
Composite integers are integers that are not prime integers.
integers since no domain element maps to a prime number.
g(x)=(x + 3) mod 26.
Then h ◦ (g ◦ f) is a function that implements the Caesar cypher as described in Example 0.3.3 (Page 3).
