ECS 20 Study Guide - Midterm Guide: Number Theory, Prime Number, Floor And Ceiling Functions
12 Oct 2018
School
Department
Course
Professor
Name:__________________________________
ID:______ ___________________________
1
ECS 20: Discrete Mathematics
Midterm
November 14, 2016
Notes:
1) Midterm is open book, open notes. No computers though…
2) You have 40 minutes, no more: We will strictly enforce this.
3) You can answer directly on these sheets (preferred), or on loose paper.
4) Please write your name at the top right of at least the first page that you turn in!
5) Please, check your work!
Part I: sets (1 question, 10 points)
1) Let A and B be two sets in a domain D. Show that
A∩B
( )
∪B∩A
( )
=A∩B
( )
∪B∩A
( )
Name:__________________________________
ID:______ ___________________________
2
Part II: functions (3 questions; each 10 points; total 30 points)
1) Let x be a real number. Solve
3x−2
⎢
⎣⎥
⎦=x
.
2) Let x be a real number. Show that
x
2
⎢
⎣
⎢⎥
⎦
⎥+x+1
2
⎢
⎣
⎢⎥
⎦
⎥=x
⎢
⎣⎥
⎦
Name:__________________________________
ID:______ ___________________________
3
Part III: Number theory (2 questions; each 10 points; total 20 points)
1) Let a, b, c be three natural numbers. Show that if b /a and c/a and gcd(b,c) = 1 then (bc) / a.
2) Show that there are no integer solutions to the equation x2-3y2 = -1
Document Summary
Notes: midterm is open book, open notes. Part i: sets (1 question, 10 points: let a and b be two sets in a domain d. show that a b. Part ii: functions (3 questions; each 10 points; total 30 points: let x be a real number. Solve 3x 2: let x be a real number. Part iii: number theory (2 questions; each 10 points; total 20 points: let a, b, c be three natural numbers. Show that if b /a and c/a and gcd(b,c) = 1 then (bc) / a: show that there are no integer solutions to the equation x2-3y2 = -1. Id:______ __________________________: show that 13 divides 3126 + 5126. Find all positive non-zero solutions of x x = x2 x floor function. Let a and b be two sets in a domain d. show that (a(cid:84) b)(cid:83)(b(cid:84) a) = (a(cid:84) b)(cid:83)(b(cid:84) a). We can use a proof by membership table.
