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Set TheoryQuestion and Answers: Page 6

Question Number 31237 Answers: 0 Comments: 3

Show that A−B = B′ ∩ A.

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{A}−\mathrm{B}\:=\:\mathrm{B}'\:\cap\:\mathrm{A}. \\ $$

Question Number 29014 Answers: 0 Comments: 3

Prove that A∪A^c =A

$${Prove}\:\:{that}\:{A}\cup{A}^{{c}} ={A} \\ $$

Question Number 28806 Answers: 1 Comments: 0

If n(A)=15 and n(B)=25, (a) What are the greatest and least values of n(AuB)? (b) What are the greatest and least value of n(AnB)? (c) Draw Venn diagrams to illustrate the four situations in (a) and (b) above

$$\mathrm{If}\:\mathrm{n}\left(\mathrm{A}\right)=\mathrm{15}\:\mathrm{and}\:\mathrm{n}\left(\mathrm{B}\right)=\mathrm{25},\: \\ $$$$\left(\mathrm{a}\right)\:\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{and}\:\mathrm{least}\:\mathrm{values}\:\mathrm{of}\:\mathrm{n}\left(\mathrm{AuB}\right)? \\ $$$$\left(\mathrm{b}\right)\:\mathrm{What}\:\mathrm{are}\:\mathrm{the}\:\mathrm{greatest}\:\mathrm{and}\:\mathrm{least}\:\mathrm{value}\:\mathrm{of}\:\mathrm{n}\left(\mathrm{AnB}\right)? \\ $$$$\left(\mathrm{c}\right)\:\mathrm{Draw}\:\mathrm{Venn}\:\mathrm{diagrams}\:\mathrm{to}\:\mathrm{illustrate}\:\mathrm{the}\:\mathrm{four}\: \\ $$$$\:\:\:\:\:\:\:\mathrm{situations}\:\mathrm{in}\:\left(\mathrm{a}\right)\:\mathrm{and}\:\left(\mathrm{b}\right)\:\mathrm{above} \\ $$

Question Number 25723 Answers: 0 Comments: 0

Given a_1 , a_2 , ..., a_n are non−negative integers and satisfy (1/2^a_1 ) + (1/2^a_2 ) + ... + (1/2^a_n ) = (1/3^a_1 ) + (2/3^a_2 ) + ... + (n/3^a_n ) = 1 If n is positive integer, find all possible solution of n

$$\mathrm{Given}\:{a}_{\mathrm{1}} ,\:{a}_{\mathrm{2}} ,\:...,\:{a}_{{n}} \:\mathrm{are}\:\mathrm{non}−\mathrm{negative} \\ $$$$\mathrm{integers}\:\mathrm{and}\:\mathrm{satisfy} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}^{{a}_{\mathrm{1}} } }\:+\:\frac{\mathrm{1}}{\mathrm{2}^{{a}_{\mathrm{2}} } }\:+\:...\:+\:\frac{\mathrm{1}}{\mathrm{2}^{{a}_{{n}} } }\:=\:\frac{\mathrm{1}}{\mathrm{3}^{{a}_{\mathrm{1}} } }\:+\:\frac{\mathrm{2}}{\mathrm{3}^{{a}_{\mathrm{2}} } }\:+\:...\:+\:\frac{{n}}{\mathrm{3}^{{a}_{{n}} } }\:=\:\mathrm{1}\: \\ $$$$\mathrm{If}\:{n}\:\mathrm{is}\:\mathrm{positive}\:\mathrm{integer},\:\mathrm{find}\:\mathrm{all}\:\mathrm{possible}\:\mathrm{solution} \\ $$$$\mathrm{of}\:{n}\: \\ $$

Question Number 24930 Answers: 1 Comments: 0

a+a=

$${a}+{a}= \\ $$

Question Number 24366 Answers: 1 Comments: 1

Given the 7-element set A = {a, b, c, d, e, f, g}, find a collection T of 3- element subsets of A such that each pair of elements from A occurs exactly in one of the subsets of T.

$$\mathrm{Given}\:\mathrm{the}\:\mathrm{7}-\mathrm{element}\:\mathrm{set}\:{A}\:=\:\left\{{a},\:{b},\:{c},\right. \\ $$$$\left.{d},\:{e},\:{f},\:{g}\right\},\:\mathrm{find}\:\mathrm{a}\:\mathrm{collection}\:{T}\:\mathrm{of}\:\mathrm{3}- \\ $$$$\mathrm{element}\:\mathrm{subsets}\:\mathrm{of}\:{A}\:\mathrm{such}\:\mathrm{that}\:\mathrm{each} \\ $$$$\mathrm{pair}\:\mathrm{of}\:\mathrm{elements}\:\mathrm{from}\:{A}\:\mathrm{occurs}\:\mathrm{exactly} \\ $$$$\mathrm{in}\:\mathrm{one}\:\mathrm{of}\:\mathrm{the}\:\mathrm{subsets}\:\mathrm{of}\:{T}. \\ $$

Question Number 22162 Answers: 1 Comments: 0

use the appropite set law to show that (A−B)∪(B−A)=(A∪B)−(A∩B)

$${use}\:{the}\:{appropite}\:{set}\:{law}\:{to}\:{show}\: \\ $$$${that} \\ $$$$\left({A}−{B}\right)\cup\left({B}−{A}\right)=\left({A}\cup{B}\right)−\left({A}\cap{B}\right) \\ $$

Question Number 22161 Answers: 0 Comments: 1

The students were asked whether they had dictionary(D) or thesau rus(T) in their room.the results showed that 650 students had dict ionary,150 did not had dictionary, 175 had a thesaurus,and 50 had neither a dictionary nor a thesaur us,fimd the number of student who (i)live in domitory ( ii)have both dictionary and thesaurus (iii)have only thesaurus

$${The}\:{students}\:{were}\:{asked}\:{whether} \\ $$$${they}\:{had}\:{dictionary}\left({D}\right)\:{or}\:{thesau} \\ $$$${rus}\left({T}\right)\:{in}\:{their}\:{room}.{the}\:{results}\: \\ $$$${showed}\:{that}\:\mathrm{650}\:{students}\:{had}\:{dict} \\ $$$${ionary},\mathrm{150}\:{did}\:{not}\:{had}\:{dictionary}, \\ $$$$\mathrm{175}\:{had}\:{a}\:{thesaurus},{and}\:\mathrm{50}\:{had} \\ $$$${neither}\:{a}\:{dictionary}\:{nor}\:{a}\:{thesaur} \\ $$$${us},{fimd}\:{the}\:{number}\:{of}\:{student}\:{who} \\ $$$$\:\:\left({i}\right){live}\:{in}\:{domitory} \\ $$$$\:\:\:\left(\:{ii}\right){have}\:{both}\:{dictionary}\:{and}\:{thesaurus} \\ $$$$\:\:\left({iii}\right){have}\:{only}\:{thesaurus} \\ $$$$ \\ $$

Question Number 21588 Answers: 0 Comments: 1

Show that if G is a finite group of even order, then G has an odd number of elements of order 2.

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{if}\:{G}\:\mathrm{is}\:\mathrm{a}\:\mathrm{finite}\:\mathrm{group}\:\mathrm{of} \\ $$$$\mathrm{even}\:\mathrm{order},\:\mathrm{then}\:{G}\:\mathrm{has}\:\mathrm{an}\:\mathrm{odd} \\ $$$$\mathrm{number}\:\mathrm{of}\:\mathrm{elements}\:\mathrm{of}\:\mathrm{order}\:\mathrm{2}. \\ $$

Question Number 19634 Answers: 1 Comments: 0

How many ordered triplets (x, y, z) of positive integer satisfy lcm(x, y) = 72, lcm(x, z) = 600 and lcm(y, z) = 900?

$$\mathrm{How}\:\mathrm{many}\:\mathrm{ordered}\:\mathrm{triplets}\:\left({x},\:{y},\:{z}\right)\:\mathrm{of} \\ $$$$\mathrm{positive}\:\mathrm{integer}\:\mathrm{satisfy}\:\mathrm{lcm}\left({x},\:{y}\right)\:=\:\mathrm{72}, \\ $$$$\mathrm{lcm}\left({x},\:{z}\right)\:=\:\mathrm{600}\:\mathrm{and}\:\mathrm{lcm}\left({y},\:{z}\right)\:=\:\mathrm{900}? \\ $$

Question Number 14028 Answers: 0 Comments: 0

Question Number 13200 Answers: 2 Comments: 0

(6)^(1/(5)^(1/(2)^(1/(√3)) ) ) = x How to write x in standard form?

$$\sqrt[{\sqrt[{\sqrt[{\sqrt{\mathrm{3}}}]{\mathrm{2}}}]{\mathrm{5}}}]{\mathrm{6}}\:=\:{x} \\ $$$$\mathrm{How}\:\mathrm{to}\:\mathrm{write}\:{x}\:\mathrm{in}\:\mathrm{standard}\:\mathrm{form}? \\ $$

Question Number 12291 Answers: 1 Comments: 0

Question Number 10000 Answers: 0 Comments: 0

An analyst was hired to survey 20 students. He reported that 6 eat eba, 5 eat amala and 7 eat semovita. 9 eat eba or amala, 12 amala or semovita and 9 eba or semovita. 3 eat all the three food and 10 eat none. After a careful analysis of these findings, the analyst was fired. why ???

$$\mathrm{An}\:\mathrm{analyst}\:\mathrm{was}\:\mathrm{hired}\:\mathrm{to}\:\mathrm{survey}\:\mathrm{20}\:\mathrm{students}. \\ $$$$\mathrm{He}\:\mathrm{reported}\:\mathrm{that}\:\mathrm{6}\:\mathrm{eat}\:\mathrm{eba},\:\mathrm{5}\:\mathrm{eat}\:\mathrm{amala}\:\mathrm{and} \\ $$$$\mathrm{7}\:\mathrm{eat}\:\mathrm{semovita}.\:\mathrm{9}\:\mathrm{eat}\:\mathrm{eba}\:\mathrm{or}\:\mathrm{amala},\:\mathrm{12}\:\mathrm{amala} \\ $$$$\mathrm{or}\:\mathrm{semovita}\:\mathrm{and}\:\mathrm{9}\:\mathrm{eba}\:\mathrm{or}\:\mathrm{semovita}.\:\mathrm{3}\:\mathrm{eat}\:\mathrm{all} \\ $$$$\mathrm{the}\:\mathrm{three}\:\mathrm{food}\:\mathrm{and}\:\mathrm{10}\:\mathrm{eat}\:\mathrm{none}.\:\mathrm{After}\:\mathrm{a}\:\mathrm{careful} \\ $$$$\mathrm{analysis}\:\mathrm{of}\:\mathrm{these}\:\mathrm{findings},\:\mathrm{the}\:\mathrm{analyst}\:\mathrm{was}\:\mathrm{fired}. \\ $$$$\mathrm{why}\:??? \\ $$

Question Number 9423 Answers: 1 Comments: 3

if x and y are two sets such that n(x) =17 , n(y)=23 and n(X∪Y) =38, find n(X∪Y).

$${if}\:{x}\:{and}\:{y}\:{are}\:{two}\:{sets}\:{such}\:{that}\:{n}\left({x}\right) \\ $$$$=\mathrm{17}\:,\:{n}\left({y}\right)=\mathrm{23}\:{and}\:{n}\left({X}\cup{Y}\right)\:=\mathrm{38}, \\ $$$$\boldsymbol{{find}}\:\boldsymbol{{n}}\left({X}\cup{Y}\right). \\ $$

Question Number 7440 Answers: 0 Comments: 7

Prove that If A, B and C are subset of the same universal set then (A − B) ∩ (A − C) = A − (B − C)

$${Prove}\:{that} \\ $$$${If}\:\:{A},\:{B}\:{and}\:{C}\:{are}\:{subset}\:{of}\:{the}\:{same}\:{universal}\:{set} \\ $$$${then}\:\left({A}\:−\:{B}\right)\:\cap\:\left({A}\:−\:{C}\right)\:=\:{A}\:−\:\left({B}\:−\:{C}\right) \\ $$

Question Number 5446 Answers: 2 Comments: 0

$$\mathrm{7} \\ $$

Question Number 5088 Answers: 0 Comments: 0

According to wikipedia, the cardonality of a set of all finite subsets of any countably infinite set is ℵ_0 . How can we prove this?

$$\mathrm{According}\:\mathrm{to}\:\mathrm{wikipedia}, \\ $$$$\mathrm{the}\:\mathrm{cardonality}\:\mathrm{of}\:\mathrm{a}\:\mathrm{set}\:\mathrm{of}\:\mathrm{all}\:{finite} \\ $$$$\mathrm{subsets}\:\mathrm{of}\:\mathrm{any}\:\mathrm{countably}\:\mathrm{infinite}\:\mathrm{set} \\ $$$$\mathrm{is}\:\aleph_{\mathrm{0}} . \\ $$$$ \\ $$$$\mathrm{How}\:\mathrm{can}\:\mathrm{we}\:\mathrm{prove}\:\mathrm{this}? \\ $$

Question Number 5080 Answers: 0 Comments: 0

N = {0, 1, 2, ...} ∣N∣ = ℵ_0 I know that: if S = {k, 0, 1, 2, ...} ∴ ∣S∣ = ∣N∣ = ℵ_0 Because you can forever pair one value from one set to another. In my question bellow I use the word ′combine′. What I mean is something like this: {1, 2, 3} + {2, 3, 4} = {1, 2, 3, 2, 3, 4} or = {1, 2, 2, 3, 3, 4} order doesn′t matter i am unsure how to write this mathematically correct My question is as follows: If we combine ℵ_0 lots of sets, with each set containing ℵ_0 values, is the total values greater than ℵ_0 ? e.g. G = N + N + ... + N ℵ_0 times G = {0, 0, ..., 1, 1, ..., 2, 2, ...} Is ∣G∣>ℵ_0 ?

$$\boldsymbol{{N}}\:=\:\left\{\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:...\right\} \\ $$$$\mid\boldsymbol{{N}}\mid\:=\:\aleph_{\mathrm{0}} \\ $$$$ \\ $$$$\mathrm{I}\:\mathrm{know}\:\mathrm{that}: \\ $$$$\mathrm{if}\:\:\boldsymbol{{S}}\:=\:\left\{{k},\:\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:...\right\} \\ $$$$\therefore\:\mid\boldsymbol{{S}}\mid\:=\:\mid\boldsymbol{{N}}\mid\:=\:\aleph_{\mathrm{0}} \\ $$$$\mathrm{Because}\:\mathrm{you}\:\mathrm{can}\:\mathrm{forever}\:\mathrm{pair}\:\mathrm{one}\:\mathrm{value} \\ $$$$\mathrm{from}\:\mathrm{one}\:\mathrm{set}\:\mathrm{to}\:\mathrm{another}. \\ $$$$ \\ $$$$\mathrm{In}\:\mathrm{my}\:\mathrm{question}\:\mathrm{bellow}\:\mathrm{I}\:\mathrm{use}\:\mathrm{the}\:\mathrm{word} \\ $$$$'{combine}'.\:\mathrm{What}\:\mathrm{I}\:\mathrm{mean}\:\mathrm{is}\:\mathrm{something} \\ $$$$\mathrm{like}\:\mathrm{this}: \\ $$$$\left\{\mathrm{1},\:\mathrm{2},\:\mathrm{3}\right\}\:+\:\left\{\mathrm{2},\:\mathrm{3},\:\mathrm{4}\right\}\:=\:\left\{\mathrm{1},\:\mathrm{2},\:\mathrm{3},\:\mathrm{2},\:\mathrm{3},\:\mathrm{4}\right\} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{or}\:=\:\left\{\mathrm{1},\:\mathrm{2},\:\mathrm{2},\:\mathrm{3},\:\mathrm{3},\:\mathrm{4}\right\} \\ $$$${order}\:{doesn}'{t}\:{matter} \\ $$$${i}\:{am}\:{unsure}\:{how}\:{to}\:{write}\:{this}\:{mathematically}\:{correct} \\ $$$$ \\ $$$$\mathrm{My}\:\mathrm{question}\:\mathrm{is}\:\mathrm{as}\:\mathrm{follows}: \\ $$$$\mathrm{If}\:\mathrm{we}\:\mathrm{combine}\:\aleph_{\mathrm{0}} \:\mathrm{lots}\:\mathrm{of}\:\mathrm{sets},\:\mathrm{with}\:\mathrm{each} \\ $$$$\mathrm{set}\:\mathrm{containing}\:\aleph_{\mathrm{0}} \:\mathrm{values},\:\mathrm{is}\:\mathrm{the}\:\mathrm{total} \\ $$$$\mathrm{values}\:\mathrm{greater}\:\mathrm{than}\:\aleph_{\mathrm{0}} \:? \\ $$$$ \\ $$$${e}.{g}. \\ $$$$\boldsymbol{{G}}\:=\:\boldsymbol{{N}}\:+\:\boldsymbol{{N}}\:+\:...\:+\:\boldsymbol{{N}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\aleph_{\mathrm{0}} \:\mathrm{times} \\ $$$$\boldsymbol{{G}}\:=\:\left\{\mathrm{0},\:\mathrm{0},\:...,\:\mathrm{1},\:\mathrm{1},\:...,\:\mathrm{2},\:\mathrm{2},\:...\right\} \\ $$$$\mathrm{Is}\:\mid\boldsymbol{{G}}\mid>\aleph_{\mathrm{0}} ? \\ $$

Question Number 4733 Answers: 0 Comments: 10

Question Number 4027 Answers: 0 Comments: 2

Let A denotes the Set of Algebraic Numbers and T the Set of Trancedental Numbers. Discuss the following: •Are A and T closed with respect to addition and multiplication ? •Are A−{0) and T closed with respect to division?

$${Let}\:\mathbb{A}\:{denotes}\:{the}\:{Set}\:{of}\:{Algebraic}\:{Numbers} \\ $$$${and}\:\:\mathbb{T}\:\:\:{the}\:{Set}\:{of}\:{Trancedental}\:{Numbers}. \\ $$$${Discuss}\:{the}\:{following}: \\ $$$$\bullet{Are}\:\mathbb{A}\:{and}\:\mathbb{T}\:\:\boldsymbol{{closed}}\:{with}\:{respect}\:{to}\:\: \\ $$$$\boldsymbol{{addition}}\:{and}\:\boldsymbol{{multiplication}}\:? \\ $$$$\bullet{Are}\:\mathbb{A}−\left\{\mathrm{0}\right)\:\:{and}\:\mathbb{T}\:\:\boldsymbol{{closed}}\:{with}\:{respect}\:{to}\:\: \\ $$$$\boldsymbol{{division}}? \\ $$

Question Number 3551 Answers: 0 Comments: 4

Prove that P set of prime numbers is countable.

$$\mathrm{Prove}\:\mathrm{that}\:\mathbb{P}\:\mathrm{set}\:\mathrm{of}\:\mathrm{prime}\:\mathrm{numbers}\:\mathrm{is} \\ $$$$\mathrm{countable}. \\ $$

Question Number 3130 Answers: 2 Comments: 0

If A,B,C and D are any four sets then (i) (A−B)∪(C−D)=^(?) (A∪C)−(B∪D) (ii) (A−B)∪(C−D)=^(?) (A∪C)−(B∩D)

$${If}\:{A},{B},{C}\:{and}\:{D}\:{are}\:{any}\:{four}\:{sets}\:{then} \\ $$$$\left({i}\right)\:\:\left({A}−{B}\right)\cup\left({C}−{D}\right)\overset{?} {=}\left({A}\cup{C}\right)−\left({B}\cup{D}\right) \\ $$$$\left({ii}\right)\:\left({A}−{B}\right)\cup\left({C}−{D}\right)\overset{?} {=}\left({A}\cup{C}\right)−\left({B}\cap{D}\right) \\ $$

Question Number 1875 Answers: 0 Comments: 0

Given that: Z={0, 1, 2, ...} all integers ≥0 R={0, 0.01, ..., 1, 1.01, ...} all reals ≥0 Prove that ∣R∣>∣Z∣

$$\mathrm{Given}\:\mathrm{that}: \\ $$$${Z}=\left\{\mathrm{0},\:\mathrm{1},\:\mathrm{2},\:...\right\}\:\mathrm{all}\:\mathrm{integers}\:\geqslant\mathrm{0} \\ $$$${R}=\left\{\mathrm{0},\:\mathrm{0}.\mathrm{01},\:...,\:\mathrm{1},\:\mathrm{1}.\mathrm{01},\:...\right\}\:\mathrm{all}\:\mathrm{reals}\:\geqslant\mathrm{0} \\ $$$$\:\mathrm{Prove}\:\mathrm{that}\:\mid{R}\mid>\mid{Z}\mid \\ $$

Question Number 1763 Answers: 1 Comments: 1

$$ \\ $$

Question Number 1744 Answers: 3 Comments: 0

If A and B are two sets and U is a universal set prove that A ⊆ B ⇒ B=A ∪ (A′ ∩ B)

$${If}\:\boldsymbol{\mathrm{A}}\:{and}\:\boldsymbol{\mathrm{B}}\:{are}\:{two}\:{sets}\:{and}\:\mathbb{U}\:{is} \\ $$$${a}\:{universal}\:{set}\:{prove}\:{that} \\ $$$$\boldsymbol{\mathrm{A}}\:\subseteq\:\boldsymbol{\mathrm{B}}\:\:\Rightarrow\:\boldsymbol{\mathrm{B}}=\boldsymbol{\mathrm{A}}\:\cup\:\left(\boldsymbol{\mathrm{A}}'\:\cap\:\boldsymbol{\mathrm{B}}\right) \\ $$

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