问题1.以下哪个是空集示例?
(i)除以5的所有偶数自然数的集合。
(ii)所有偶数素数的集合。
(iii){x：x 2 –2 = 0且x是有理数}。
(iv){x：x是自然数，x <8并同时x> 12}。
(v){x：x是任意两条平行线的共同点}。
解决方案：
(i) All the numbers ending with 0. Except 0 is divisible by 5 and are even natural number.
Therefore, it is not an example of empty set.
(ii) 2 is a prime number and is even, and it is the only prime which is even.
Therefore, this not an example of the empty set.
(iii) x2 – 2 = 0, x2 = 2, x = ± √2 ∈ N. There is not natural number whose square is 2.
Therefore, it is an example of empty set.
(iv) There is no natural number less than 8 and greater than 12.
Therefore, it is an example of the empty set.
(v) No two parallel lines intersect at each other.
Therefore, it is an example of empty set.
问题2.以下哪些集合是有限的，哪些是无限的?
(i)平面中的一组同心圆。
(ii)英文字母集。
(iii){x∈N：x> 5}
(iv){x∈N：x <200}
(v){x∈Z：x <5}
(vi){x∈R：0
解决方案：
(i) Infinite concentric circles can be drawn in a plane.
Therefore, it is an infinite set.
(ii) There are just 26 letters in English Alphabets.
Therefore, it is finite set.
(iii) It is an infinite set because, natural numbers greater than 5 is infinite.
(iv) It is a finite set. Since, natural numbers start from 1 and there are 199 numbers less than 200.
Therefore, it is a finite set.
(v) It is an infinite set. Because integers less than 5 are infinite.
Therefore, it is an infinite set.
(vi) It is an infinite set. Because between two real numbers, there are infinite real numbers.
问题3.以下哪些集合相等?
(i)A = {1,2,3}
(ii)B = {x∈R：x 2 –2x + 1 = 0}
(iii)C =(1、2、2、3}
(iv)D = {x∈R：x 3 – 6x 2 + 11x – 6 = 0}
解决方案：
A set is said to be equal with another set if all elements of both the sets are equal and same.
A = {1, 2, 3}
B ={x ∈ R: x2 – 2x+1=0}
x2 – 2x+1 = 0
(x–1)2 = 0
Therefore, x = 1.
B = {1}
C= {1, 2, 2, 3}
In sets we do not repeat elements hence C can be written as {1, 2, 3}
D = {x ∈ R: x3 – 6x2+11x – 6 = 0}
For x = 1, x2–2x+1=0
= (1)3–6(1)2+11(1)–6
= 1–6+11–6
= 0
For x =2,
= (2)3–6(2)2+11(2)–6
= 8–24+22–6
= 0
For x =3,
= (3)3 – 6(3)2+11(3)–6
= 27–54+33–6
= 0
Therefore, D = {1, 2, 3}
Hence, the set A, C and D are equal.
问题4.以下集合是否相等?
A = {x：x是reap单词中的字母}，
B = {x：x是单词paper中的字母}，
C = {x：x是单词rope中的字母}。
解决方案：
For A
Letters in word reap
A ={R, E, A, P} = {A, E, P, R}
For B
Letters in word paper
B = {P, A, E, R} = {A, E, P, R}
For C
Letters in word rope
C = {R, O, P, E} = {E, O, P, R}.
Set A = Set B
Because every element of set A is present in set B
But Set C is not equal to either of them because all elements are not present.
问题5.从下面给出的集合中，将等效集合配对：
A ＝ {1、2、3}，B ＝ {t，p，q，r，s}，C ＝ {α，β，γ}，D ＝ {a，e，i，o，u}。
解决方案：
Equivalent set are different from equal sets, Equivalent sets are those which have equal number of elements they do not have to be same.
A = {1, 2, 3}
Number of elements = 3
B = {t, p, q, r, s}
Number of elements = 5
C = {α, β, γ}
Number of elements = 3
D = {a, e, i, o, u}
Number of elements = 5
Therefore, Set A is equivalent with Set C.
Set B is equivalent with Set D.
问题6.以下几组对相等吗?说明原因。
(i)A = {2，3}，B = {x：x是x 2 + 5x + 6 = 0}的解
(ii)A = {x：x是单词“ WOLF”的字母}
B = {x：x是单词“ Follow”的字母}}
解决方案：
(i) A = {2, 3}
B = x2 + 5x + 6 = 0
x2 + 3x + 2x + 6 = 0
x(x+3) + 2(x+3) = 0
(x+3) (x+2) = 0
x = -2 and -3
= {–2, –3}
Since, A and B do not have exactly same elements hence they are not equal.
(ii) Every letter in WOLF
A = {W, O, L, F} = {F, L, O, W}
Every letter in FOLLOW
B = {F, O, L, W} = {F, L, O, W}
Therefore, A and B have same number of elements which are exactly same, hence they are equal sets.
问题7.从下面给出的集合中，选择相等的集合和等同的集合。
A = {0，a}，B = {1,2,3,4}，C = {4，8，12}，
D = {3，1，2，4}，E = {1，0}，F = {8，4，12}，
G = {1，5，7，11}，H = {a，b}
解决方案：
A = {0, a}
B = {1, 2, 3, 4}
C = {4, 8, 12}
D = {3, 1, 2, 4} = {1, 2, 3, 4}
E = {1, 0}
F = {8, 4, 12} = {4, 8, 12}
G = {1, 5, 7, 11}
H = {a, b}
Equivalent sets:
i. A, E, H (They have exactly two elements in them)
ii. B, D, G (They have exactly four elements in them)
iii. C, F (They have exactly three elements in them)
Equal sets:
i. B, D (all of them have exactly the same elements, Hence they are equal)
ii. C, F (all of them have exactly the same elements, Hence they are equal)
问题8.以下哪些集合相等?
A = {x：x∈N，x <3}
B = {1，2}，C = {3，1}
D = {x：x∈N，x为奇数，x <5}
E = {1,2,1,1}
F = {1，1，3}
解决方案：
A = {1, 2}
B = {1, 2}
C = {3, 1}
D = {1, 3} (since, the odd natural numbers less than 5 are 1 and 3)
E = {1, 2} (since, repetition is not allowed)
F = {1, 3} (since, repetition is not allowed)
Therefore, Sets A, B and E are equal.
C, D and F are equal.
问题9。证明拼写“ CATARACT”所需的字母集和拼写“ TRACT”所需的字母集是相等的。
解决方案：
For “CATARACT”
Distinct letters are
{C, A, T, R} = {A, C, R, T}
For “TRACT”
Distinct letters are
{T, R, A, C} = {A, C, R, T}
As it is seen that the letters need to spell cataract is equal to set of letters need to spell tract.
Therefore, the two sets are equal.