Discrete Math 6A

Max Welling

Recap

1. Proposition: statement that is true or false.

2. Logical operators: NOT, AND, OR, XOR, ,

3. Compound proposition: a new proposition by combining old ones using operators.

4. Logical equivalences: pq always true.

5. Predicate: property of a statement with variables (>3).

6. Quantifier: “there is” , “for all”. These turn statements with variables into propositions.

1.4 Nested Quantifiers

Nested Quantifier: Quantifier that appears within the scope of another quantifier.

Examples: x,y real numbers.)0( yxyx

))()(( zyxzyxzyx x,y,z real. (associative law for addition).

)0()0()0( xyyxyx x,y real.

-For all x and for all y if x is positive and y is negative then their product must be negative.-The product of a positive and a negative real number is negative.

Translate this sentence into a logical expressions.“If a person is female and is a parent, then she is someone’s mother.”F(x) is “x is female”, P(x) is “x is someone’s parent”, M(x,y), “x is the mother of y”

),())()((( yxMxPxFyx

1.4 Order of QuantifiersImportant: The order in which quantifiers occur can be very important! i.e not always true.

Example: x,y real.

Is this proposition true?

)0()0( yxyxyxxy

left: There is a y such that for all x (x+y=0). (F)right: For all x there is a y such that x+y=0 (T)Proposition is therefore False.

What do we learn? The order is important!

),(),( yxPyxyxPxy

However the following compound propositions are always true. Left and right are equivalent.

),(),(

),(),(

yxPxyyxPyx

yxxPyyxyPx

1.4 Nested QuantifiersTip: You can think of expression with quantifiers as executing “loops” in a computer program:

Example:

First loop over y and and for every y loop over x.For every value of y, check if P(x,y) is true for all x.If you found one, the proposition must be true.

),( yxPxy

Examples on white board

1.6 SetsSets are the most fundamental discrete structure in mathematics. A lot is based on it!We now develop the formal theory of sets.This basically boils down to a lot of definitions.

Set: Unordered collection of “objects”

We denote sets with capitals. The objects are also called “members”or elements. Thus set A is said to contain its elements.

Example: V={a,e,i,o,u} (vowels in English) brackets are used to define the set. {.}

C = all students subscribed to Math6A in spring 2003.

1.6 SetsWell-known sets in math: N = {0,1,2,3,...} Z = {...,-2,-1,0,1,2,...} Z+ = {1,2,3,...} Q = {p/q | p in Z, q in Z, q is not 0} R = {x | x is a real number}.

{,...} is used to indicate the the rest of the sequence once it’s clear how to proceed {1,2,3,...}

set builder notation: {x | conditions(x) }.This could be read as “all x such that the conditions hold true”.

Definition: Two sets are equal iff they contain the same elements.

Example: {1 2 2 3 3 3 3 4 6 } = {1 2 3 4 6} = {6 4 1 3 2}

( ) ( ) A B iff x x A x Bnotation for “x is anelement of B”.

1.6 Sets

Subset: A is a subset of B iff every element of A is an element of B.

( ) A B iff x x A x B

notation for: “A is a subset of B”

Example: all even integers are a subset of all integers. all Math6A students are a subset of all students.

1.6 Sets

Venn diagrams

U

AB a

U: universal set: set of all objects under consideration.

1.6 Sets

Empty set: Set that contains no elements. {}

Proposition: For any nonempty set S :

Attention: { }

( ) ( )i S ii S S

Is this true?

( ) S iff x x x S(i)

The premises: is false, and thus the implication is true. x

(ii) ( ) S S iff x x S x SClearly the right is true, so the left must be true.

1.6 Sets.

Proper Subset: A is a proper subset of B iff A is a subset of B and A is not equal to B

( ) ( ) A B iff A B A B

Sets may contain other sets as members!

Example: {0, {a},{b}, {a,b}} = {x|x is a subset of {a,b}}Note “a” is not “{a}”.

Cardinality: The cardinality of A is the number of distinct elements in A: |A|. Also: S is finite in this case. Infinite set: a set that is not finite. (e.g. all integers, real numbers).

countable uncountable

1.6 Sets

Power set: The power set P(A) of A is the set of all subsets of A.

Example: P({1,2,3}) = {O, {1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}

P(O)={O}

Ordered n-Tuple: (a1,a2,...,an) is an ordered collection that has a1 is its first element , a2 as its second etc.

Note: where sets are unordered this structure has ordered elements.(a,b) and (b,a) are not the same unless a=b.In general (a1,a2,a3,...an) = (b1,b2,b3,...bn) iff a1=b1, a2=b2, ...

1| ({ ,..., }) | 2nnP x x

1.6 sets

Cartesian product: The set of all ordered pairs (a,b) where a in A and b in B.

{( , ) | }A B a b a A b B

Note: we have mapped two sets to a new set.

Example: A={1,2,3} B={a,b}

AXB = {(1,a),(1,b),(2,a),(2,b),(3,a),(3,b)}.

Note that order in (.) is important, but order in {.} is not important.

AXB is not the same as BXA ! (unless A=B).

Examples on white board.