Step1 of1 A
Suppose there is an integers a such that,
(1:qu +rl
=d42+r2
where q,q2,r, and r2 are integers, 0S1; < d,0$ r2 <d, and r, at r2.
By interchanging the labels for I; and r2 , and assume that r2 > 1;.
Then.
dmrw=a-n>0
because both i; and r, are le
Step 10 of 11 A
Consider both the real numbers negative as -2, -3.
The absolute value of the sum is,
l-2 +(-s)| = l-Sl
= 5
The sum of the absolute values is,
|2| +|3| = 2+3
= 5
In this case also, the absolute value of the sum is equal to the sum of the ab
Step 2 of 3 A
b)
The first part of the statement represents R , in the second part by considering Ftto be negative,
we get the third part as 3J_ R .
3J_ R being equal to x which is equal to s, sis also negative.
For any real numbers, it sis negative, then
Step1 of 2 A
The objective is to fill in the blanks with a variable or variables using the following statement:
Is there a real number whose square is 1?"
(a)
Let x be a real number.
The square of the number x is represented by x.
The square of a number i
Step1 of1 A
(a) Given any two real number, there is a real numb er in between. Now we have to
write this statement using variable. Blanks part are given in colors.
Given anyr two real number a and b , there is a real number e such that I: is
between a and
Practice Test 3, M325K, 12/04/2014
PRINTED NAME:
No books, notes, calculators, or telephones are allowed.
Every problem is worth an equal number of points.
You must show your work; answers without substantiation do not count.
Answers must appear in the bo
Discrete Math M325K
1
Set Theory
Dr. Berg
The Basics
The founder of abstract set theory, Georg Cantor (1845-1918), described a set as a
collection into a whole M of definite and separate objects of our intuition or thought.
These objects are called the el
Exam 2
Name_
1)
a)
M325K
EID_
Dr. Berg
Fall 09
Give counterexamples to these assertions.
For all real numbers x and y, xy = x y.
b)
For all real numbers x and y, x y = x y .
2)
Use a proof by contradiction to prove that the sum of any rational number and
Discrete Math M325K
1
Functions
Dr. Berg
The Basics
Now we study relations and, more specifically, functions.
Definition
A relation R from a set A to a set B is a subset of A B . If (a, b) R we write
aRb and say that a is related to b by R. The set A is c
Step1of3 A
(a) All nonzero real numbers have a reciprocal
Step 2 of 3 A
(b) For all nonzero real numbers r , there is a reciprocal for 3'.
Comment
Step 3 of 3 A
(c) For all nonzero real numbers r , there is areal number s such that g is
reciprocal of r.
Step 2 of 3 A
b)
The first part of the given statement introduces x. We need to mention the attributes of x in the
second part.
Thus,
There is a real number rsuch that the product of rwith every number is the same number.
Comment
Step 3 of 3 A
C)
In the g
Step1 of 2 A
The objective is to prove that (J5 + J5) is irrational by using contradiction method.
Assume that this is not true. i.e., J5 + J3 is rational.
By the definition of rational,
Ji+ J5 = B, where p and q are any integers with q at 0.
q
Squaring o
Step 3 of 3 A
The above result shows that 9 I 02.
Thus, 9:2 and 9|(a2 3). from supposition
By the properties of divisibility, it can be written as:
9 I (a= -(a= 4)
9 | 3
This is a contradiction.
So our supposition is wrong.
Thus, 9 does not divide (a2 3)f
Step1 of 5 A
The negation of a statement of the type If a then b is given as follows:
If not b, then not b
Comment
Step 2 of 5 A
(a)
To prove the following statement:
If for all prime numbers p > 2, x + y = 2P has no positive integer solution, then for an
Step1 of2 A
(a) N1=pl+l= 2+1=3
Nz=pl-p,+l=2-3+l=?
hi3=3a~p2-p3+l=2-3-5+l=3l
N=p1-p, .223.p+1= 2-3-5-?+1=211
N, =p1-pgcp3'ppp5 +l= 2-3-5-T-ll+l=23ll
N=p1-pg-p3-p-p5-p+l= 2-3-5-T-ll-l3+l=3003l
Step 2 of2 A
(b) We know that the prime factors of n can be chec
Step1 of1 A
Let us consider the following statement
For all equations E, if E is quadratic then E has at most two real solutions.
The statement can be rewritten as shown below.
(a) All quadratic equations have at the most two real solutions.
(b) Every qua
Step 2 of 3 A
b)
We are given R in the given statement.
Now the blank must represent the attributes of r, which according to the given Universal
Existential Statement are positive and square root of Ft.
Thus,
For every positive number 9 there is a positiv
Step1of1 A
The example for d is not prime d | n2 , but of does net need to be divisible by a
Let d = 9 (9 is not a prime number)
Then, 9 | 62 = 36 but 9 does not divide 6
Step 2 of 5 A
(b) Every square has four sides.
Step 3 of 5 A
(c) If an object is a square, then it has four sides.
Step 4 of 5 A
(d) If J is a square, then J has four sides.
Comment
Step 5 of 5 A
(e) For all squares J , J has four sides.
Step 1 of 2 A
Consider that the sequence cfw_an is defined as follows:
a =2"_+('_l)_l, for all integers n 2 0_ . (1)
' 4
The objective is to find an explicit formula for an that includes floor notation.
Suppose nis even.
Then n = 2k , for some integer k.
Step1 of 2 A
At first proof the solution by contradiction.
Suppose not.
This is, suppose n is another integer satisfying the below conditions that n2 + 2, _3is prime.
Now factorise the n2 +2n~3-
n2 +2n3=(nl)(n+3).
But n>0,andso nl>land n+3)!-
Hence n2 + 2
Step1 of 2 A
Given that: The reciprocal of any positive real number is positive. Now we have to
write this statement using variable.
(a) Given an!r positive real number 3* , the reciprocal of I; is positive.
(b) For an!r real number 3* , if r is positive,