Set A formal collection of objects, which can be anything May be finite or infinite Can be defined by: – Listing the elements – Drawing a picture – Writing a rule defining what is inside Operations –    ’ –  = – Cartesian Product: A  B = { (x, y) | x  A and y  B } Think of picking food from a menu: how many possible meals? The empty set is a subset of every set. It’s unique.

Sets of strings Begin with an alphabet Σ Σ n = all words with n “letters” Σ * = all words of any length, including length 0 * in general means “0 or more instances of” – 0* means all words containing any number of 0’s only – (0 + 11)* means any concatentations of “0” and “11” “language” = any set of words from Σ * Example: Let’s say Σ = { a, b }. – L = { ε, a, b, aa, ab, ba, bb } is a language with 7 words. This language happens to be Σ 0  Σ 1  Σ 2. Example: set of legal identifiers

Working with sets Boolean algebras Characteristic value of a set Bit vectors – Representation – Operations – Use in programming language Example: Vacation program – Bit masks

Properties Set theory and propositional logic are both boolean algebras. – What do we mean by “an algebra” ? Definition of a boolean algebra: a mathematical system with values 0 and 1 and operators + and * having these properties: – Closure, Commutativity, Distributivity, identity element, inverse Logic and set theory are “subclasses” of boolean algebra. They inherit these properties.

Properties (2) What is the practical consequence? Expressions involving set operations are analogous to boolean expressions. So, to verify that A  (B  C)  (A  B)  (A  C), we can do so by converting it to the corresponding propositional formula, and seeing if it’s a tautology. a  (b  c)  (a  b)  (a  c) In general, another way to show X  Y is to pick an arbitrary x  X and show why it must be in Y. – To show X = Y is done in two parts: X  Y and Y  X.

Characteristic value Often, a set consists of just whole numbers. – E.g. { 1, 4, 6 } In this case, it’s convenient to refer to the set by a single numerical value: the set’s characteristic value. The value is the sum of all 2 i where i is a number in the set. – E.g. { 1, 4, 6 }  The binary representation of this number shows the set! This is called a bit vector

Logic operations Examples AND AND Try the same examples with “OR” and “XOR” Observations: – What happens when you AND with a 1? With a 0? – What about OR’ing with a 1 versus a 0? – What about XOR? How would we … – Make the rightmost three bits all 1’s? All 0’s?

Operations Inside the computer, set operations on bit vectors are very fast. Notation in C, C++ and Java: | & ^ ~ Let’s look at x = { 1, 4, 6 } and y = { 3, 4, 5 }. How would you handle subset? what31… x111 y111 x | y11111 x & y1 x ^ y1111 ~x Blank cells are zero.

Binary number Value is based on powers of two: This number equals = 173 Interesting relationship between unary ~ and – ~n = – (n + 1) …

Shorthand Occasionally we need to write a large number in binary. Better to do so as “hexadecimal” – a shorthand that compresses 4 bits in one symbol. Notation: begin with “0x” followed by, usually, 8 digits. Each digit is: 0-9, a-f a = ten = “1010”d = thirteen = “1101” b = eleven = “1011”e = fourteen = “1110” c = twelve = “1100”f = fifteen = “1111” Example: What does 0x7ffc0 look like in binary?

Shift operations Two directions – Shift left (<<) basically means to multiply by a power of 2. – Shift right (>>) essentially divides by a power of 2 and eliminates the remainder. For example 1 << 4 = 2 4 = 16 5 << 3 = 5 * 2 3 = >> 1 = 13 / 2 1 = 6

Practical use Let’s suppose we want to use an integer to hold 31 boolean values, representing the days of a month. We’ll use bit positions 1-31 only. Start with sept = 0, clearing all bits. To turn on a certain bit, we need to “or with 1” sept = sept | (1 << 29) To turn off a bit, we need to “and with 0” sept = sept & ~(1 << 29) How would we determine value of just 29 th bit? Multiple bits? …

Mask Often we want to inspect/modify more than 1 bit, so we need to create an operand that looks like: – Ones in middle: – Zeros in middle: How? Two ways – Can subtract powers of 2 – Use only bitwise operations

Examples ~0 … ~0 << 5 … ~(~0 << 5)… ~(~0 << 5) << 4… And you can invert one more time if needed. How could we write this as a difference of powers of 2 ?

Bit field Store more than one value in a variable, to save space. Example: date as month-day-year – How many bits do we need for each? date = 0 date |= month << 16 date |= day << 11 date |= year MonthDayYear 4511