Unit 2 review Discrete Math 226

prime number definition -

n is prime <-> ∀ positive integers r and s , if n=rs, then r=1 and s=n or s=1 and r =n Note : n must be greater than 1 Special Notes : 1 is not a prime Every integer greater than 1 can be prime or composite In simple terms, a prime # only factors are 1 and itself ! Example primes: 2,3,5,7,11,13

partition of a set -

A finite or infinite collection of nonempty sets {A1, A2, A3 ...} is a partition of a set A if, and only if, 1. A is the union of all the Ai 2. The sets A1, A2, A3,... are mutually disjoint.

sequence -

A sequence is a function whose domain is either all the integers between two given integers or all the integers greater than or equal to a given integer.

getting proofs started -

Showing the starting point and the conclusion to be shown

disproving existence or existential statements

Disproving Existential Statements A statement of the form ∃x ∈ D, P(x) is false if and only if P( ) x is false for all x ∈ D. To disprove this kind of statement, we need to show the for all x ∈ D, P(x) is false.

disproving universal statements

Disproving you only have to show one example which disproves - Called a counterexample Example: Disprove for all real numbers a and b if a<b = then a^2 < b^2 Counterexample: Let a=-2 and b=-1 -2 < -1 but -2^2 > -1^2 Cause we using real #s gotta consider negatives

Closed form for sum of terms -

Theorem 5.2.2 Sum of the First n Integers For all integers n ≥ 1, 1 + 2 +···+ n = n(n + 1)/2 .

Set Relations -

Theorem 6.2.1 Some Subset Relations 1. Inclusion of Intersection: For all sets A and B, (a) A ∩ B ⊆ A and (b) A ∩ B ⊆ B. 2. Inclusion in Union: For all sets A and B, (a) A ⊆ A ∪ B and (b) B ⊆ A ∪ B. 3. Transitive Property of Subsets: For all sets A, B, and C, if A ⊆ B and B ⊆ C, then A ⊆ C.

Combinations -

(n chooses r) = n!/r ! · (n − r )!

Sets, proper subsets -

A is a proper subset of B ⇔ (1) A⊆ B, and (2) there is at least one element in B that is not in A.

definition of rational numbers - ratio of integers

A real number r is rational iff can be expressed as a quotient of two integers with non zero denominator. A real number not rational is irrational r is rational <-> ∃ integers an and b such that r=a/b and b doesn't equal zero Examples: 10/3 is a rational 2/0 is not a rational Theorem 4.2.1 Every integer is a rational Theorem 4.22 The sum of two rationals is rational

Cartesian product -

Cartesian Products Let A1 = {x, y}, A2 = {1, 2, 3}, and A3 = {a, b}. a. Find A1 × A2. b. Find (A1 × A2) × A3. Solution a. A1 × A2 = {(x, 1), (x, 2), (x, 3), (y, 1), (y, 2), (y, 3)} b. The Cartesian product of A1 and A2 is a set, so it may be used as one of the sets making up another Cartesian product. This is the case for (A1 × A2) × A3. (A1 × A2) × A3 = {(u, v)| u ∈ A1 × A2 and v ∈ A3} by definition of Cartesian product = {((x, 1), a), ((x, 2), a), ((x, 3), a), ((y, 1), a), ((y, 2), a), ((y, 3), a), ((x, 1), b), ((x, 2), b), ((x, 3), b), ((y, 1), b), ((y, 2), b), ((y, 3), b)}

even/odd parity -

Consecutive integers have opposite parity An even number is a number which has a remainder of 0 upon division by 2, while an odd number is a number which has a remainder of 1 upon division by 2 Thus, the set of integers can be partitioned into two sets based on parity: the set of even (or parity 0) integers, and the set of odd (or parity 1) integers.

Factorial notations n! 0! -

For each positive integer n, the quantity n factorial denoted n!, is defined to be the product of all the integers from 1 to n: n! = n ·(n − 1)··· 3·2·1. Zero factorial, denoted 0!, is defined to be 1: 0! = 1.

Real Number intervals open, closed and without bounds infinite -

Given real numbers a and b with a ≤ b: (a, b) = {x ∈ R | a < x < b} [a, b]={x ∈ R | a ≤ x ≤ b} (a, b]={x ∈ R | a < x ≤ b} [a, b) = {x ∈ R | a ≤ x < b}. The symbols ∞ and −∞ are used to indicate intervals that are unbounded either on the right or on the left: (a,∞) = {x ∈ R | x > a} [a,∞) = {x ∈ R | x ≥ a} (−∞, b) = {x ∈ R | x < b} [−∞, b) = {x ∈ R | x ≤ b}.

Geometric Sequence -

In a geometric sequence, each term is obtained from the preceding one by multiplying by a constant factor. If the first term is 1 and the constant factor is r, then the sequence is 1,r,r 2,r 3,...,r n,.... The sum of the first n terms of this sequence is given by the formula, see image

Testing Whether One Set Is a Subset of Another

Let A = {1} and B = {1,{1}}. a. Is A ⊆ B? b. If so, is A a proper subset of B? Solution a. Because A = {1}, A has only one element, namely the symbol 1. This element is also one of the elements in set B. Hence every element in A is in B, and so A ⊆ B. b. B has two distinct elements, the symbol 1 and the set {1} whose only element is 1. Since 1 = {1}, the set {1} is not an element of A, and so there is an element of B that is not an element of A. Hence A is a proper subset of B.

Order n-tuple and Ordered triple

Let n be a positive integer and let x1, x2,..., xn be (not necessarily distinct) elements. The ordered n-tuple, (x1, x2,..., xn), consists of x1, x2,..., xn together with the ordering: first x1, then x2, and so forth up to xn. An ordered 2-tuple is called an ordered pair, and an ordered 3-tuple is called an ordered triple. Two ordered n-tuples (x1, x2,..., xn) and (y1, y2,..., yn) are equal if, and only if, x1 = y1, x2 = y2,..., xn = yn. Symbolically: (x1, x2,..., xn) = (y1, y2,..., yn) ⇔ x1 = y1, x2 = y2,..., xn = yn. In particular, (a, b) = (c, d) ⇔ a = c and b = d.

Power set - set of all subsets -

Power Set of a Set P({x, y}) = {∅,{x},{y},{x, y}}

Basis step -

Prove P(1) or whatever is the lower limit

Inductive Step -

Prove P(k+1)

Ordered n tuples example

See example

Change of a variable -

See image

Dummy variables -

See image

Mutually disjoint sets / Pairwise disjoint sets

Sets A1, A2, A3 ... are mutually disjoint (or pairwise disjoint or nonoverlapping) if, and only if, no two sets Ai and Aj with distinct subscripts have any elements in common. More precisely, for all i, j = 1, 2, 3,... Ai ∩ Aj = ∅ whenever i = j. Mutually Disjoint Sets a. Let A1 = {3, 5}, A2 = {1, 4, 6}, and A3 = {2}. Are A1, A2, and A3 mutually disjoint? Solution a. Yes. A1 and A2 have no elements in common, A1 and A3 have no elements in common, and A2 and A3 have no elements in common.

Sets, set notation -

Sets can be related to each other. If one set is "inside" another set, it is called a "subset". Suppose A = { 1, 2, 3 } and B = { 1, 2, 3, 4, 5, 6 }. Then A is a subset of B, since everything in A is also in B. This relationship is written as: A is a subset B A⊂B That sideways-U thing is the subset symbol, and is pronounced "is a subset of". To show something is not a subset, you draw a slash through the subset symbol, so the following: B is not a subset of A ...is pronounced as "B is not a subset of A".

Set Difference (relative complement) -

The difference of B minus A (or relative complement of A in B), denoted B − A, is the set of all elements that are in B and not A.

Intersection -

The intersection of A and B, denoted A ∩ B, is the set of all elements that are common to both A and B.

Direct Proof

The steps are as follow: 1. Express the statement in the form as the above statement. 2. Start the proof by assuming x is an element of M for which the hypothesis P(x) is true. 3. Show that Q(x) is true by definitions, previous result and the rules for logical inference. Example: Prove the following statement by the method of direct proof: Suppose a and b are even integers. Prove that the sum and difference of a and b are divisible by 2. Let a = 2m and b = 2n for some integer m and n. Sum of a and b = 2m + 2n = 2(m + n). Difference of a and b = 2m - 2n = 2(m-n). Since m and n are even integers, m + n and m - n are also even. Hence, 2(m + n) and 2(m - n) are multiples of 2. Therefore, for some even integers a and b, their sum and difference are divisible by 2.

Union -

The union of A and B, denoted A ∪ B, is the set of all elements that are in at least one of A or B.

Ordered pair -

Two ordered pairs (a, b) and (c, d) are equal if, and only if, a = c and b = d. Symbolically: (a, b) = (c, d) means that a = c and b = d. Example Ordered Pairs a. Is (1, 2) = (2, 1)? a. No. By definition of equality of ordered pairs, (1, 2) = (2.1) if, and only if, 1 = 2 and 2 = 1.

Disjoint Sets -

Two sets are called disjoint if, and only if, they have no elements in common. Symbolically: A and B are disjoint ⇔ A ∩ B = ∅. Disjoint Sets Example Let A = {1, 3, 5} and B = {2, 4, 6}. Are A and B disjoint? Solution Yes. By inspection A and B have no elements in common, or, in other words, {1, 3, 5}∩{2, 4, 6}=∅.

proving universal statements

Universal Statement - which says that a concept is true for a set of elements There are 2 methods - generalization from a generic particular and exhaustion Exhaustion example: if n is even and 4<=n <= 26 then n can be written as 2 prime numbers, 4=2+2, 6=3+3, 8 = 3+5, 16=15+11,.... 26=17+9 Generalization from a generic, you suppose that x is a particular but arbitrarily chosen element of the set D that makes the hypothesis P(X) true

Sets, subsets -

We begin by rewriting what it means for a set A to be a subset of a set B as a formal universal conditional statement: A ⊆ B ⇔ ∀x, if x ∈ A then x ∈ B.

Empty Set -

We call it the empty set (or null set) and denote it by the symbol ∅. Thus {1, 3}∩{2, 4}=∅ and {x ∈ R | x 2 = −1}=∅.

Summation notation - index, lower limit, upper limit -

first value = lower limit last value = upper limit

corollary -

is a statement whose truth could be immediately deduced from a theorem that has already been proved Example : Corollary 4.2.3 the double of a rational is a rational - Its proven by theorem 4.22 The Sum of 2 rationals is a rational, because a double of a rational is just the sum of itself + itself

composite number definition -

positive integers r and s such that n=rs and 1<r<n and 1<s<n Note : n must be greater than 1 Example composites are 4,6,8,9,10,12

n /d equals

quotient Example 77 div 11 = 7

n mod d equals

remainder Example 77 mod 10 = 7

Arithmetic sequence -

see image

Set identities -

see image

standard factor form, prime factorization -

see image

Power Set - empty set -

℘(∅) { ∅ } (the set whose only element is the empty set). The empty set ∅ is a subset of every set, so ∅ is in every powerset.

A quick reference of the symbols used and their meanings follows -

∃ - there exists ∈ - belongs to ∋ - such that ∀ - For all values of

Even integer definition -

∃ an integer k , n=2k Example: Is 6m+14n even/odd/either? 2(3m+7n) so its even - consider your 3m+7n = k

odd integer definition -

∃ an integer k, n =2k+1 Example: Is 12mn+7 even/odd/either (12mn+ 6) + 1 2(mn+3) + 1 - so odd you consider mn+3 = k

proving existence - constructive proof vs non constructive proof

Existential Statement - which says that something exists, or is true for certain elements Some examples of existential statements are - There exists a natural number n, such that n x n = 36 There exists an integer z, such that z^2=25 There is at least one number n, belonging to a set of Natural numbers, such that a x n = a Constructive Proof: Proving Existential Statements - ∃x ∈ D such that Q(x) is true if, and only if, Q(x) is true for at least one x in D Example: Prove there exists an even integer that can be written in 2 ways as a sum of prime numbers Solution : Let n=10 then 10=5+5=3+7 and 3,5, and 7 are all primes A nonconstructive proof of existence involves showing either a) that the existence of a value of x that makes Q(x) true is guaranteed by an axiom or a previously proved theorem or b) that the assumption that there is no such x leads to a contradiction. The disadvantage of a nonconstructive proof is that it may give virtually no clue about where or how x may be found.

Expanded form of a sequence -

See image

Product notation -

See image

Sequence example

See image

Summation Notation Example

See image

Transformation of a Summation Notation Example

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Transforming of a variable for Summation Example

See image

Complement in a Set -

The complement of A, denoted Ac , is the set of all elements in U that are not in A. Find the complement of B in U B = { 1, 2, 4, 6} U = {1, 2, 4, 6, 7, 8, 9 } Complement of B in U = { 7, 8, 9}

definition of divisibility - several equivalent terms

if n and d are integers and d does not equal 0 n is divisible by d if and only if n equals d times some integer Equivalent terms are n is a multiple of d d is a factor of n d is a divisor of n d divides n notation d | n reads d divides n d cannot equal zero d | n <-> ∃ integer k such than n=dk Theorem 2.1. For all positive integers a and b, if a|b then a ≤ b. Theorem 2.2. The only divisors of 1 are 1 and −1.

zero product property

if neither of two real numbers is zero, then their product is also not zero AB=0 if either A or B is zero