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CMSC250 Midterm 1 Notes, Study notes of Discrete Mathematics

University of Maryland Discrete Mathematics

midterm notes about logic and probability

Typology: Study notes

2018/2019

Uploaded on 04/30/2019

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CMSC250 Midterm #1

Propositional Logic

• Propositional symbols: building blocks of propositional logic – boxes that hold value of 1 or 0

• Three basic operations

• Conjuction (AND, ^) = binary operator

p q p ∧ q

FFF

T F F

F T F

TTT

• Disjunction (OR) = binary operator

p q p ∨ q

F F F

F T T

T F T

T T T

• Negation (NOT, ~, ¬) = unary operator

p~p

F T

T F

• Binary connectives

•→ if then

p q pq

F F T

F T T

T F F

T T T

•↔ if and only if

p q p↔q

F F T

F T F

Discover Study notes of Discrete Mathematics University of Maryland

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CMSC250 Midterm # Propositional Logic

Propositional symbols: building blocks of propositional logic – boxes that hold value of 1 or 0
Three basic operations
- Conjuction (AND, ^) = binary operator p q^ p^ ∧^ q F F F T F F F T F T T T
Disjunction (OR) = binary operator p q p ∨ q F F F F T T T F T T T T
Negation (NOT, ~, ¬) = unary operator p ~p F T T F
Binary connectives
→ if then p q pq F F T F T T T F F T T T
↔ if and only if p q p↔q F F T F T F

T F F

T T T

False implies anything
Logical equivalence compares two statements while if and only if forms statements
- p ∧ q ≡ q ∧ p
De Morgan’s Law – conjunctions flipped to disjunctions and negation operator distributed across terms - ~ (p ∧ q) ≡ ~p ∨ ~q
Contrapositive – p q ≡ ~q ~p
More logical equivalences
- ∼ a ∨ b ≡ a b
- a ↔ b ≡ (∼ a ∨ b) ∧ (∼ b ∨ a)
satisfiability
- truth table – takes 2n^ rows for n propositional symbols Circuits

A out 0 1 1 0

inverter – corresponds to the negation operation

AND gate – corresponds to conjunction p q^ r 0 0 0 0 1 0 1 0 0 1 1 1

n-bit adder
- consists of n-1 full adders and 2n-1 half-adders Quantifiers
Ǝ – there exists
2 C 6 F- for all
vacuously true: if no element exists in first quantifier, it doesn’t make the second part not true but rather just vacuously true
nesting quantifiers – order of quantifiers matters
(∃ x ∈ ℕ)(∀ y ∈ ℕ)[x < y]
To define even, odd, and prime
Even(x) ↔ ((∃ y ∈ Z) [x = 2y])
Prime(x) ↔ [y divides x → ((y=1) ∨ (y=x)]
Odd(x) ↔ ∼Even(x)
Existential statement - (Ǝ x ϵ D) [P(x)]
To prove true – need to find one instance
To prove false – must scan entire domain to show there is not x it holds for
Universal statement - (2 C 6 F^ x^ ϵ^ D) [P(x)]
To prove true – must show for entire domain
To prove false – must provide a counterexample
Negated quantifiers ∼Ǝ is the same as Ɐ
Dense: a set is dense if between any two elements of it there exists another element of it Set Theory
Set: collection of distinct objects
To show a member of a set say “y ∈ Z”
To represent sets
enumeration (“curly brace” notation)

■ A = {-1,2,8}

Through a property ■ S1 = {x| x ∈ ℝ ∧ x ≥ 100}
Subset: if every element in A is an element of B, A is a subset of B (A ⊆ B)
B is a superset of A
If there is one element that is an element of B but not A, then A is a proper subset of B (A ⊂ B)
Empty set, denoted either ∅ or {} – unique set with no elements
Union between A and B (A ∪ B): set of elements that belong either to A or to B
Intersection between A and B (A ∩ B): set of elements that belong to both A and B
Mutually disjoint: if there is no intersection between two sets A, B (A ∩ B = ∅)
Rule for union
Absolute complement of A (Ac): set of all elements that do not belong to A
Relative complement of A given B (B-A or B\A): set of all elements in B but not in A
Powerset: set of all subsets of A (includes null set)
Cardinality(|A|)
For a finite set – number of elements
For an infinite set – positive infinity
Partition - {A 1 , A^2 , …, An } is a partition of A if
A (^) i and A (^) j are mutually disjoint, for i, j ∈ {1, 2, …, n} with i ≠ j.
A 1 ∪ A 2 ∪ ⋯ ∪ An = A
Cartesian product (A X B): set of ordered pairs of elements of A and B Functions
Most basic representation – arrow diagrams
Also can represent a function as
f: A ↦ B ■ A – domain ■ B – co-domain
Every element of the domain should map to some co-domain element

Proof by contradiction – assume what we want to prove is false
- Eventually leads to a contradiction and can prove what you assumed is false
Modular arithmetic – a ≡ b (mod m) or a is congruent to b mod m
- m | (a-b)
Properties of equivalence
- If a 1 ≡^ b^1 (mod m) and a 2 ≡^ b 2 (mod m), then (a 1 + a 2 )^ ≡^ (b 1 + b 2 ) (mod m)
  - If a 1 ≡ b 1 (mod m) and a 2 ≡ b 2 (mod m), then (a 1 * a 2 ) ≡ (b 1 * b 2 ) (mod m)
Proof by contraposition – proving the contrapositive may be easier (pq is the same as ~q ~p)
Unique Factorization Theorem: every natural number greater than or equal to 2 can be uniquely factored into a product of prime numbers p 1 , p2 ,…, pn :

Sequences and Series

Sequence: function from the naturals to the complex numbers
- Can be written in three forms ■ Outlining terms ■ “closed form” formula ■ Recursive formula – must have base cases
Recursion vs closed formula
Computation ■ Recursion leads to a fast dynamic program ■ Closed form is faster but numerical issues arise
Rate of growth ■ Recursion gives no hint as how big F n is ■ Closed form gives approximate rate of growth
Summation notation – sum of sequence a 1 ,a 2 ,…,ak is denoted as
The i = 1 represents the index of term with which the sum begins
The k represents the index of last term to be included in the sum

Properties of summation notation
Product notation: product of sequence a 1 ,a 2 ,…,a (^) k is denoted as
Properties of product notation
Series: sum of all elements of an infinite sequence
Partial sum(Sn ): sum ranging from the first up to (and including) the nth^ term of a sequence
Arithmetic sequence: defined by the formula an = a 0 + n*d
Geometric sequence: defined by the formula m n*a (^0) Weak Induction
To prove a proposition P(n), want to prove two separate things
P(0) is true
For any k ≥ 0, if P is true for k, it then must be true for k + 1
Three different parts to a weak induction proof
Inductive base: prove that P(0) is true
Inductive hypothesis: assume P(k) for k ≥ 0
Inductive step: prove that if P(k) holds, then P(k+1) holds
Use weak induction for
Coin problems – usually have multiple cases
Summation problems
Inequalities
For two variable proofs – explicitly say at the beginning of the proof which variable you will be performing a proof by induction on Strong Induction
To prove P(n), want to prove two separate things
P(0), P(1), …, P(a) are true
For a k ≥ a, P(0) and P(1) and … and P(a) P(k+1)

CMSC250 Midterm 1 Notes, Study notes of Discrete Mathematics

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T F F

T T T

■ A = {-1,2,8}