问题1:以下哪个集合?证明我们的答案是正确的。
(i)以字母J开头的一年中所有月份的集合。
(ii)印度十位最有才华的作家的收藏。
(iii)一支由11名世界最佳板球运动员组成的团队。
(iv)班上所有男孩的收藏。
(v)收集少于100的所有自然数。
(vi)作家Munshi Prem Chand所写的小说集。
(vii)所有偶数整数的集合。
(viii)本章中的问题集。
(ix)世界上最危险的动物的集合。
解决方案:
(i) All months that begins with the letter J is a well-defined collection as one can easily identify whether a month begins with J or not.
Therefore, this collection is a set.
(ii) Ten most talented writers of India is not a well-defined collection as one writer’s talent cannotbe measured and it vary for person to person.
Therefore, this collection is not a set.
(iii) A team of eleven best cricket batsmen of the world is not a well-defined collection because one person can consider a batsman as best whereas other person may not consider it. There’s no measuring unit.
Therefore, this collection is not a set.
(iv) This collection is a well-defined collection as you can easily identify whether a boy belongs to your class or not.
Therefore, this collection is a set.
(v) The collection of all natural numbers less than 100 is a well-defined collection because it’s possible to decide whether a number is less than 100 or not.
Therefore, this collection is a set.
(vi) A collection of novels written by the writer Munshi Prem Chand is a well-defined collection because one can definitely identify whether a book is written by Munshi Prem Chand or not.
Therefore, this collection is a set.
(vii) This collection is a well-defined collection because one can definitely identify which is an even integer and which is not.
Therefore, this collection is a set.
(viii) This collection is a well-defined collection because one can definitely identify whether a question belongs to this chapter or not.
Therefore, this collection is a set.
(ix) This collection of most dangerous animals of the world is not a well-defined collection because the criteria for determining the dangerousness of an animal varies from person to person.
Therefore, this collection is not a set.
问题2:让A = {1,2,3,4,5,6}。在空格中插入适当的符号∈或::
(i)5…A
(ii)8…A
(iii)0…A
(iv)4…A
(v)2…A
(vi)10…A
解决方案:
(i) 5 ∈ A
(ii) 8 ∉ A
(iii) 0 ∉ A
(iv) 4 ∈ A
(v) 2 ∈ A
(vi) 10 ∉ A
问题3:以花名表形式编写以下几组:
(i)A = {x:x是一个整数,并且–3
(ii)B = {x:x是小于6的自然数}。
(iii)C = {x:x是两位自然数,因此其总和为8}
(iv)D = {x:x是一个质数,它是60的因子。
(v)E =三角词中所有字母的集合。
(vi)F =单词BETTER中所有字母的集合。
解决方案:
(i) A = {–2, –1, 0, 1, 2, 3, 4, 5, 6}
(ii) B = {1, 2, 3, 4, 5}
(iii) C = {80, 71, 62, 53, 44, 26, 35, 17}
(iv) Prime factorization of 60 is
60 = 22 × 31 × 51
D = {2, 3, 5}
(v) E = {Y, E, M, N, O, G, I, R, T}
(vi) F = {R, T, E, B}
问题4:以set-builder形式编写以下集合:
(i)(3、6、9、12)
(ii){2、4、8、16、32}
(iii){5,25,125,625}
(iv){2,4,6…}
(v){1、4、9…100}
解决方案:
(i) {3, 6, 9, 12}
3 = 3*1, 6 = 3*2, 9 = 3*3, 12 = 3*4.
∴ Set builder form is {x: x = 3n, n ∈ N and 1 ≤ n ≤ 4}
(ii) {2, 4, 8, 16, 32}
We know that all the numbers can be written as following 2 = 21, 4 = 22, 8 = 23, 16 = 24, 32 = 25.
∴ Set builder form is {x: x = 2n, n ∈ N and 1 ≤ n ≤ 5}
(iii) {5, 25, 125, 625}
We know that all the numbers can be written as following 5 = 51, 25 = 52, 125 = 53, and 625 = 54.
∴ Set builder form is {x: x = 5n, n ∈ N and 1 ≤ n ≤ 4}
(iv) {2, 4, 6 …}
Given set is a set of all even natural numbers.
∴ Set builder form is {x: x is an even natural number}
(v) {1, 4, 9, 16, 25 … 100}
We know that all the numbers can be written as following, 1 = 12, 4 = 22, 9 = 32 …100 = 102.
∴ Set builder form is {x: x = n2, n ∈ N and 1 ≤ n ≤ 10}
问题5:列出以下集合的所有元素:
(i)A = {x:x是一个奇数自然数}
(ii)B = {x:x是一个整数,-1/2
(ⅲ)C = {X:X是一个整数,X 2≤4}
(iv)D = {x:x是单词“ LOYAL”中的字母}
(v)E = {x:x是一年中没有31天的月份}
(vi)F = {x:x是英语字母的辅音,后跟k}。
解决方案:
(i) A= {1, 3, 5, 7, 9 …}
(ii) B = {0,1,2,3,4}
(iii) We know that
(–1)2 = 1 ≤ 4
(–2)2 = 4 ≤ 4
(–3)2 = 9 > 4
02 = 0 ≤ 4
12 = 1 ≤ 4
22 = 4 ≤ 4
32 = 9 > 4
∴ C = {–2, –1, 0, 1, 2}
(iv) D = {L, O, Y, A}
(v) E = {February, April, June, September, November}
(vi) F = {b, c, d, f, g, h, j}
问题6:将名册形式左边的每个集合与集合构建者形式中描述的右边的相同集合匹配:
(i) {1, 2, 3, 6} | (a) {x: x is a prime number and a divisor of 6} |
(ii) {2, 3} | (b) {x: x is an odd natural number less than 10} |
(iii) {M, A, T, H, E, I, C, S} | (c) {x: x is natural number and divisor of 6} |
(iv) {1, 3, 5, 7, 9} | (d) {x: x is a letter of the word MATHEMATICS} |
解决方案:
(i) All the elements given in this set are natural numbers as well as the divisors of 6.
Therefore, (i) matches with (c).
(ii) We know that 2 and 3 are prime numbers. They are also the divisors of 6.
Therefore, (ii) matches with (a).
(iii) This set contains all the letters from the word MATHEMATICS.
Therefore, (iii) matches with (d).
(iv) This set contains odd natural numbers which are smaller than 10.
Therefore, (iv) matches with (b).