







Some questions about sets 
20191215 

From M.Azzi: Hello,
(I)  In set theory can a given set contain both elements and subsets, as "elements", as in :
A = {1,2,{},3,{{3}},8}
If yes :
1  then, is A = 6 ?
2  if the empty set is a subset of every set,
2. 1. does {} = {{}}, {{{}}} etc? , and if the is true what are the respective cardinals of the latter three? (0,1,1?).
2 . 2. Why isn't {1} equal to {{},1}? and why should these two be equal without having the same cardinality?
Sorry if my questions are not well expressed.
Thank you for the great service you provide. Answered by Harley Weston. 





Combinations of cities 
20191203 

From Oliver: Hi!
I'm looking to find out how many combinations (non repeating) there are for 6 cities.
If we name the cities A to F, possible combinations would include;
A.
A, B.
B.
A, B, C.
A, C.
B, C.
C.
and so on.
Thank you! Answered by Penny Nom. 





nC0 + nC1 + nC2 + .... + nCn = 2^n 
20180219 

From bristal: (QQ) Prove, nC0 + nC1 + nC2 + .... + nCn = 2^n. Answered by Penny Nom. 





Subsets 
20160626 

From Kats: How Many sub sets are in set k={6,7,3} Answered by Penny Nom. 





Countable and uncountable sets? 
20160115 

From wilson: what are the countable and uncountable sets? Answered by Penny Nom. 





The number of possible musical notes using an nkey instrument 
20150504 

From Farihin: Lets say that i have keys, and each key is for notes of a musical instrument,
So i wanted to find out the number of notes i can get for a certain number keys,
of course in the form of an equation. Notes can use as many keys, it can use 1, or 2, or 3, or even 100.
Notes in real life is not as such, but ignore reality.
I tried doing this but i can't seem to find a formula for it.
For example, i have 4 keys, say A, B, C, and D.
so, for notes that uses one key are 4, which is A, B, C, and D themselves.
for notes that uses two keys are 6,
AB, AC, AD, BC, BD and CD.
for notes that uses three keys are 4,
ABC, ABD, ACD and BCD.
lastly for notes that uses all four keys is 1, ABCD.
So, the total will be 4+6+4+1=15#
The nth term for the first equation is n, the second is [(n^2)n]/2
the third and the fourth, i don't know but the final answer should be like,
n + [(n^2)n]/2 + [3rd] + [4th]
Sorry for the long question though... Answered by Penny Nom. 





A question in set theory 
20150225 

From Jared: If a set A={1,2,3} and set B={ {}, 1}
Can B be a subset of A? Since every Set contains an {} ? Answered by Robert Dawson and Claude Tardif. 





Overlapping sets 
20140523 

From daniel: motors inc manufactured 325 cars with automatic transmission,216 with power steering ,and 89 with both these options. How cars were manufactured if every car has at least one option? Answered by Penny Nom. 





Equivalent sets 
20120913 

From asif: show that (1,1)~(1,1) or give its counter example Answered by Harley Weston. 





The power set of A 
20120324 

From rashdin: Can you find a set A, A=4 and define a bijective function between A and P(A)? Answered by Penny Nom. 





Properties of real numbers applied to subsets 
20120201 

From Mark: Hello 
The questions that I have for you is do the properties of real numbers (such as the associative, commutative, identity, inverse, and distributive law) apply to ALL the subsets of real numbers? In other words, do all those properties work for the Natural Numbers? The Whole Numbers? And so on and so forth. I understand that they are all real numbers, but for instance: the identity is whenever you add zero to a number, you get that number back. But does that work with, say, with only the odd numbers? Zero isn't odd so can that property actually apply to JUST the odd numbers? Any consideration would be greatly appreciated! Answered by Robert Dawson. 





Cardinality of infinite sets 
20090901 

From Brian: I was reading an answer to a question on your site regarding infinite sets (http://mathcentral.uregina.ca/QQ/database/QQ.09.01/carlos1.html), and I think they may have got the answer wrong.
I his example, he claims that the set of real numbers BETWEEN 0 AND 1 is larger than the set of positive integers.
Please correct me if I am wrong, but I believe those two sets are  pardon the expression  equally infinite. For any integer, there is a corresponding real number between 0 and 1, and vice versa.
For instance, using the decimal as a "mirror", you can create a mirror image of any real number between 0 and 1 as an integer (i.e. 0.1234 gets mirrored as the integer 4321  I could write it out algebraically, if you want, but you get my point)
Am I wrong?
Thanks,
Brian Answered by Victoria West. 





Subsets 
20090616 

From Tracy: Suppose C is the subset of D and D is the subset of C.
If n(c)=5, find n(D)
What other relationship exists between sets C and D? Answered by Penny Nom. 





The axiom of choice and constructibe sets 
20090410 

From sydney: The axiom of choice asserts the existence of certain sets, but does not construct the set. What does "construct" mean here? For example, does it require showing the existence and uniqueness of some function yielding the set? In general, what does it mean to require the existence of a mathematical object be tied to a construction of it? Answered by Claude Tardif. 





Infinite sets and infinite limits 
20090306 

From Justin: Hello, I know I have asked a similar question before but I was just wondering if set theory applies to the lim x>0, y=1/x=infinity and if so, what type of infinity would it be? Thanks a lot for your help with this question!
Regards,
Justin Answered by Robert Dawson and Harley Weston. 





(a x b) intersect (b x a) 
20090108 

From sean: is it possible to have two sets such that
n((a X b) intersect (b X a) =3 Answered by Harley Weston. 





A union B and A intersect B 
20090107 

From Jim: Suppose A and B are sets and (A union B) = (A intersect B). Is it true that A=B. Answered by Penny Nom. 





The empty set 
20080929 

From wahab: Why a null set is called a set?
the definition of set includes that a set is a collection of well defined objects
But a null set is having no value. Answered by Harley Weston. 





Subsets of a set 
20071030 

From Snehal: 1. Let an denote the number of subsets of f{1,2, 3.... n}including the
empty set and the set itself.)
a) Show an = 2an1
b) Guess a formula for the value of an and use induction to prove you are
right Answered by Stephen La Rocque. 





Countable and uncountable sets 
20070724 

From Mac: Hi, i tried to read few webpages related to the countably infinite and uncountable sets.
Even i read few questions from this forum.
But i am not convinced with this explanation. If you have any good book that
explains this in layman term, please redirect me to that.
1) Can you please explain what is the difference between these too ?
2) How could you say set of Natural number and set of even numbers are countably
infinite ?
N={1,2,3,...} and Even= {2,4,6,...}
When an element in the even set is some 2n, we will map it to 'n'.So
now we have a bigger number(2n) right ?
Sorry, i didn't understand that.
...
Can you please help me out to understand that ? Answered by Harley Weston. 





Equality of sets 
20070723 

From Mac: Hi, I learnt set theory recently. My teacher and few of the weblink actually give different
definition for basic set. Can you please solve this ?
My teacher says, {1,2,3} and {1,1,2,3} is also set.
But in this link http://library.thinkquest.org/C0126820/setsubset.html it says,
"A set has no duplicate elements. An element is either a member of a set or not. It cannot be in the set twice."
and "{1, 2, 3} is the same as the set {1, 3, 2, 3, 1}"
My question is,
1. Whether duplicates allowed in the set or not ?
2. Even if the duplicates are allowed, {1,2,3} and {1,1,2,2,3,3} are same or not ? Answered by Penny Nom and Harley Weston. 





The empty set is a subset of every set 
20061114 

From Narayana: The empty set is a subset of every set Answered by Stephen La Rocque and Penny Nom. 





Onequarter of all 3subsets of the integers 1,2,3....,m contain the integer 5 
20061009 

From Hina: If onequarter of all 3subsets of the integers 1,2,3....,m contain the integer 5, determine the value of m. Answered by Steve La Rocque and Claude Tardif. 





Brackets and more brackets 
20060829 

From Michelle: Feeling stupid asking but it's been awhile ... {{{2}}} ...what is this really saying ....are the outer brackets = null? Answered by Stephen La Rocque. 





Marking out a circle 
20060628 

From Peter: given a straight line. how do i work out the off sets ( at right angles) at several intermediate points. to set out a 5.0m arc that has a 18.0m radius. Answered by Stephen La Rocque and Penny Nom. 





The cartesian product of a countably infinite collection of countably infinite sets 
20060325 

From Geetha: Is the cartesian product of a countably infinite collection of countably infinite sets countable infinite? Answered by Penny Nom. 





A countably infinite collection of countably infinite sets 
20050226 

From Feroz: Suppose a set can be divided into a countably infinite number of countably infinite sets.Then can the original set be considered as a countably infinite set? Answered by Penny Nom. 





B={A,{A}} 
20040920 

From Muhammad: Let A be a set and let B = {A,{A}}.
(a) Explain the elements of set B (with some example)
(b) Prove that A is not a subset of B. Answered by Penny. 





Equivalent sets 
20040306 

From A student: If A=(1,2,3,4,...) and B=(5,10,15,20,...), is A equivalent to B. Why or Why not ? Answered by Penny Nom. 





Sets 
20040127 

From Susan: My child has the following problems to solve, and we are puzzled.
1. Compare the subset symbols to the inequality symbols of less than or greater than.
2. If A, B & C are sets such that A has 47 elements, B has 32 elements, and C is a proper subset of B, what can you say about the number of elements in the following sets: A U B? A intersect B? B U C? and B intersect C? Answered by Penny Nom. 





A problem with sets 
20040120 

From Jason:
Given that the universal set S is the set of all sports fans, and
F={xx is a football fan}
B={xx is a basketball fan}
H={xx is a hockey fan}
a)Describe (F^B)' (f intersect b)' in words
b)Draw a Venn Diagram and shade the region that represents the set of football fans or both basketball and hockey fans.
Answered by Penny Nom. 





What is larger than infinity? 
20030112 

From Dana: What is larger than infinity? Answered by Claude Tardif and Harley Weston. 





Combinations of 1,2,3,...,10 
20021127 

From Gord: If I had the numbers from 110 how many different combinations would i have.....would it be 100....since that is 10 squared. Answered by Penny Nom. 





Two problems 
20021014 

From Eva:
a) How many different equivalence relations can be defined on the set X={a,b,c,d}? b)Show that 6 divides the product of any 3 consecutive integers. I know it is true that 6 divides the product of any 3 consecutive integers. However, i have problem showing the proof. Answered by Leeanne Boehm and Penny Nom. 





Sets and elements 
20020822 

From Dianne: I want to know why its okay to say that, for example, 6 is an element of the set of integers, but you get counted off for saying that the set of 6 is an element of the set of integers. How come? Answered by Judi McDonald. 





Can a infinite set be smaller than another infinite set? 
20011129 

From Carlos: Can a infinite set be smaller than another infinite set? If so why? Answered by Chris Fisher and Penny Nom. 





Cardinality of sets 
20011119 

From Tania:
 Show that the cardinality of P(X) (the power set of X) is equal to the cardinality of the set of all functions from X into {0,1}.
 Show that (the cardinality of the natural numbers set) N = NxNxN.
 Show that the cardinality of the set of prime numbers is the same as the cardinality of N+
Answered by Walter Whiteley. 





Subsets of a countably infinite set 
20011114 

From Tania: How could I show (and explain to my son) that any countably infinite set has uncontably many infinite subsets of which any two have only a finite number of elements in common? Answered by Claude Tardif. 





Subsets of the natural numbers 
20010130 

From Christina: How do I explain why the set of natural numbers (N) cannot be equivalent to one of its finite subsets? Answered by Penny Nom. 





Derfs, Enajs and Sivads 
20010107 

From John and Norman: All Derfs are Enajs. Onethird of all Enajs are Derfs. Half of all Sivads are Enajs. One Sivad is a Derf. Eight Sivads are Enajs. The number of Enajs is 90. How many Enajs are neither Derf nor Sivad? Answered by Penny Nom. 





Infinite sets 
20000412 

From Brian Provost: Here's the deal: There are an infinite amount of integers (1,2,3...). Agreed? There are an infinite amount of even integers, too (2,4,6...). Agreed? By convention, infinity equals infinity. Yet common sense tells us there are obviously more integers than there are even integers. Prove this to be true mathematically. Answered by Harley Weston. 





100% on two tests 
20000201 

From Craig and Chelsea Bruzzone: A class of 35 students took a math test and a science test. 12 students got 100% on the math test. 9 students got 100% on the science test. There were 19 students who made less than 100% on both tests. How many students made 100% on both tests? Answered by Penny Nom. 

