CS 205: Introduction to Discrete Structures
Final
Date 12/21/16
Professor: Naftaly Minsky
Each of the first five questions is worth 15 points, while question 6 is worth 25
points.
Please justify each
Homework 3 CS 205 Fall 2014
Due Sunday, September 5, 11pm
(Our goal is to be able to take up solutions to this homework next week,
before the test. Hence this homework is somewhat shorter/easier.
Depe
Homework 5
Due SUNDAY, November 2nd, 2014
A. In each of the following cases, either prove the equality for all m>=2, a, b or find a counter-example with the smallest m possible.
1. a%m = b%m if a=b (m
HOMEWORK 4
Due *SATURDAY NIGHT 11 pm*, October 25th, 2014
The purpose of these excercises is to give you practice in developing
mathematical proofs. So please read textbook Section 1.7 and beginning o
5*.[more challangening] Suppose you are given a third inductive denition of string-like things @ over alphabet :
1. is in @ ;
2. is in @ for every letter in the alphabet ;
3. x.y is in @ whenever x an
CS 198:205 Fall 2014 (Professor A. Borgida)
Homework 6 Part I
DUE: SATURDAY, November 15th, 2014 at 11:00pm
The rst part of the assignment, concerning induction can be completed now. The second part,
f: A->B
domain_f
codomain_f
single-valued: for an a in A, there is unique f(a) (NOT solutions to x^2=1
total: for each a in A there is a value f(a) (NOT division_of_3, by 0)
rng(f) = cfw_y | exists x
Facts to remember about INTEGERS (Z). UNIVERSE=Z
* aithmetic ops are associative, distributive, commutative
* if a=b then a # c = b # c (for /, c not 0)
* if a<b then a # c < b # c for +,-
* if a<b th
TO PROVE: 1*1! + 2*2! +. + n*n! = (n+1)! - 1 for positive integer n
>STEP 1. Write predicate P(n) = "1*1! + 2*2! +. + n*n! = (n+1)! - 1"
To prove 'IF n in N+ THEN P(n)'
[ERROR: P(n) is a logical as
INTEGERS: DEFINITIONS, FUNCTIONS , ALGORITHMS AND APPLICATIONS
(In the notes below, I will use underscore _ to indicate a subscript, and caret ^ to indicate a superscript. So b_j is really bj and b^n
The textbooks chapter on Sets follows the one on Logic, and so relies on its
notation and results.
This document is a brief introduction to sets, which reects my lectures and
which does not rely on th
"Structural induction".
A set S of objects is defined inductively/recursively via
i) one or more base cases that belong to S
ii) one or more recursive rules where "composite" elements are
added to S
Notes on Strings and Structural Recursion
198:205
Magnus Halldorsson, Ann Yasuhara, Alexander Borgida
November 8, 2014
Let be a nite, non-empty set of symbols, such as cfw_0, 1 or cfw_a, b, c.
1
Denit
Computer Science Department - Rutgers University
Spring 2018
CS 205: Homework 2 - If I Did It
16:198:205
Questions
1) Verify the following identities (distribution laws) via truth table or by giving a
Computer Science Department - Rutgers University
Spring 2018
CS 205: Homework Assignment 1
16:198:205
Complete and turn in to the best of your abilities the following problems due in class on Monday F
Computer Science Department - Rutgers University
CS 205: Things Worth Knowing for Midterm I
Spring 2018
16:198:205
Basic Math
Calculating Factorials, Calculating Exponentials
Rules of Logarithms
Th
Computer Science Department - Rutgers University
Spring 2018
CS 205: Practice Problems for Midterm
16:198:205
Prove by induction that for all n 1,
1 + 2 + 3 + . . . + (n 1) + n =
1
(n + 1)n.
2
(1)
Wh