问题1.以下哪项是空集的示例?
i)可被2整除的奇数自然数集
解决方案：
Yes, this is a null set because three is no odd natural number divisible by 2.
Note: A set which does not contain any element is called the null set.
ii)偶数质数集
解决方案：
No, this is not a null set because 2 is an even number which is prime.
iii){x：x是自然数，x <5和x> 7}
解决方案：
Yes, this is a null set because three is no natural number which is less than 5 and greater than 7.
iv){y：y是任意两条平行线的共同点}
解决方案：
Yes, this is a null set because two parallel lines have no points in common because they do not intersect.
问题2.以下哪些集合是有限的或无限的?
i)一年中的几个月
解决方案：
This is a finite set because there is only 12 months in a year.
Note: A set which is empty or consists of a definite number of elements is called finite otherwise, the set is called infinite.
ii){1,2,3，…}
解决方案：
This is an infinite set because there will always be a new number if we add 1 to the previous number, or we can say that it is a set of natural numbers and there are infinite natural numbers.
iii){1、2、3，…，99、100}
解决方案：
This is a finite set because there is only 100 numbers in the set.
iv)大于100的正整数集
解决方案：
This is an infinite set because there are infinite positive integers greater than 100 which can be generated by adding 1 to the previous number.
v)小于99的素数集
解决方案：
This is a finite set because prime numbers which are less than 99 are finite.
问题3：说明以下每个集合是有限的还是无限的：
i)平行于x轴的线组
解决方案：
Infinite. We can draw infinite parallel lines with respect to the x-axis.
ii)英文字母中的字母集
解决方案：
Finite. There are only 26 letters in the English alphabet.
iii)这组数字是5的倍数
解决方案：
Infinite. There are infinite numbers which are multiple of 5 namely {5, 10, 15 ——-}
iv)生活在地球上的那组动物
解决方案：
Finite. The animals Living on the earth can be counted they are not infinite.
v)穿过原点(0，0)的一组圆
解决方案：
Infinite. We can draw infinite number of circles passing through the origin with different radius.
问题4.在下面，说明A = B是否：
i)A = {a，b，c，d} B = {d，c，b，a}
解决方案：
Yes. Every element of A is also an element of B and every element of B is also an element of A namely {a, b, c, d}.
Note: Two sets A and B are said to be equal if they have exactly the same elements, and we write A = B. Otherwise, the sets are said to be unequal, and we write A ≠ B. Order in which elements appear does not matter.
ii)A = {4，8，12，16} B = {8，4，16，18}
解决方案：
No. 12 is an element that is present in A but not in B and similarly 18 is an element present in B not in A.
iii)A = {2，4，6，8，10} B = {x：x为正偶数且x≤10}
解决方案：
Yes. If we define set B it can be written like this {2, 4, 6, 8, 10} and therefore every element of A is also an element of B and every element of B is also an element of A.
iv)A = {x：x是10的倍数，B = {10，15，20，25，30，…}
解决方案：
No. If we define B we can clearly see that {-40, -30, -20, -10, 0}. All these numbers are also multiples of 10, and they are not in set B. Hence A ≠ B.
问题5.以下几对相等吗?说明原因
i)A = {2，3}，B = {x：x是x 2 + 5x + 6 = 0的解}
解决方案：
No.
By solving the equation x2 + 5x + 6,
x2 + 5x + 6 = 0
x2 + 2x + 3x + 6 = 0
(x + 2)(x + 3) = 0
x = -2, -3
Now it is clear that B can be defined as {-2, -3}.
Now -2, -3 are not in A and also 2, 3 is not in set B. Hence A ≠ B.
ii)A = {x：x是单词FOLLOW}中的字母B = {y：y是单词WOLF}中的字母
解决方案：
Yes.
It is clear that A can be defined as {F, O, L, W} an if we define B, {F, O, L, W}. Now it is clear that every element of A is also an element of B and every element of B is also an element of A namely {W, O, L, F}.
问题6.从下面给出的集合中选择相等的集合：
A = {2，4，8，12}，B = {1,2,3,4}，C = {4，8，12，14}，D = {3，1，4，2} E = { –1，1}，F = {0，a}，G = {1，–1}，H = {0，1}
解决方案：
B = D & E = G.
Note: Two sets A and B are said to be equal if they have exactly the same elements.
For A, 8∈ A, but 8∉ B,8∉ B,8∉ D,8∉ E, 8∉ F,8∉ G,8∉ H,
Hence, A is not equal to B, D, E, F, G, H.
Also, 2∈ A, but 2∉ C Hence A, ≠ C.
For B, 2∈ B, but 2∉ C,2∉ E,2∉ F,2∉ G, 2∉ H.
Hence, B is not equal to C, E, F, G, H.
Also, Every element of B can be found in D namely {1, 2, 3, 4} and vice-versa is also true. Hence B = D.
For C, 14∈ C, but 14∉ D,14∉ E,14∉ F,14∉ G, 14∉ H.
Hence C is not equal to D, E, F, G, H.
For D, 2∈ D, but 12∉ E,2∉ F,2∉ G, 2∉ H.
Hence D is not equal to E, F, G, H.
For E, -1∈ E, but -1∉ F, -1∉ H.
Hence E is not equal to F, H.
Also, Every element of E can be found in G namely {-1, 1} and vice-versa is also true. Hence E = G.
For F, 0∈ F, but 0∉ G, 0∉ H.
Hence F is not equal to G, H.
For G, -1∈ G, but -1∉ H.
Hence G is not equal to H.
So, we can observe that only B = D & E = G.