问题1.如果(x / 3 + 1，y – 2/3)=(5/3，1/3)，则求出x和y的值。
解决方案：
We know that,
If two ordered pairs are equal, then their corresponding first elements and second elements are equal.
We are given that the pairs (x/3 + 1, y – 2/3) and (5/3, 1/3) are equal, so the corresponding elements should also be equal.
So we have, (x/3 + 1) = 5/3 and (y – 2/3) = 1/3
On solving both the equations, we get –
x/3 + 1 = 5/3 and y – 2/3 = 1/3
x/3 = 5/3 – 1 and y = 1/3 + 2/3
x/3 = 2/3 and y = 3/3
x = 2 and y = 1
Therefore, x = 2 and y = 1
问题2.如果集合A有3个元素，集合B = {3，4，5}，则在(A×B)中找到元素的数目。
解决方案：
Given, number of elements of set A = 3
The elements of set B are 3, 4, and 5.
So, number of elements of set B = 3
Then, number of elements in A×B = (Number of elements in A) × (Number of elements in B)
= 3 × 3 = 9
Therefore, number of elements in A×B is 9.
问题3.如果G = {7，8}并且H = {5，4，2}，则求出G×H和H×G。
解决方案：
Given, G = {7, 8} and H = {5, 4, 2}
The cartesian product of two non-empty sets P × Q is the set of all ordered pairs of elements from P and Q, i.e.,
P × Q = {(p, q) : p ∈ P, q ∈ Q}
So, G x H = {(7, 5), (7, 4), (7, 2), (8, 5), (8, 4), (8, 2)}
and H x G = {(5, 7), (5, 8), (4, 7), (4, 8), (2, 7), (2, 8)}
问题4.陈述以下每个陈述是对还是错。如果该语句为假，则正确重写给定的语句。
(i)如果P = {m，n}并且Q = {n，m}，则P×Q = {(m，n)，(n，m)}。
(ii)如果A和B是非空集合，则A×B是一个有序对(x，y)的非空集合，使得x∈A和y∈B。
(iii)如果A = {1，2}，B = {3，4}，则A×(B∩)=∅。
解决方案：
(i) The given statement is False.
The correct statement is: If P = {m, n} and Q = {n, m}, then P x Q = { (m, m), n), (n, m), (n, n) }
(ii) The given statement is true.
(iii) The given statement is true.
问题5.如果A = {–1，1}，则找到A×A×A。
解决方案：
A × A × A for a non-empty set A is given by –
A × A × A = {(a, b, c) : a, b, c ∈ A}, where (a, b, c) is called an ordered triplet
Here, given A = {–1, 1},
So, A × A × A = {(-1, -1,-1), (-1, -1, 1), (-1, 1, -1), (-1, 1, 1), (1, -1, -1), (1,-1, 1), (1, 1,-1), (1, 1, 1)}
问题6.如果A×B = {(a，x)，(a，y)，(b，x)，(b，y)}。查找A和B。
解决方案：
Given,
A x B = { (a, x), (a, y), (b, x), (b, y) }
Since, the cartesian product of two non-empty sets P × Q is given by –
P × Q = {(p, q) : p ∈ P, q ∈ Q}
So, A is the set of all first elements and B is the set of all second elements.
Therefore, A = {a, b} and B = {x, y}
问题7.让A = {1，2}，B = {1,2,3,4}，C = {5，6}和D = {5，6，7，8}。验证
(i)A×(B×C)=(A×B)×(A×C)。
(ii)A×C是B×D的子集。
解决方案：
Given, A= {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}
(i) To verify: A × (B ∩ C) = (A × B) ∩ (A × C)
Since B ∩ C= {1,2, 3,4} ∩ {5,6} = ∅
Thus, L.H.S.= A × (B ∩ C) = A × ∅ = ∅
Now,
A x B = { (1, 1), (1, 2), (1, 3), (1, 4), (2, 1), (2, 2), (2, 3), (2, 4) }
A x C = { (1, 5), (1, 6), (2, 5), (2, 6) }
Thus, R.H.S. = (A × B) ∩ (A × C) = ∅
Therefore, L.H.S. = R.H.S
Hence, verified.
(ii) To verify: A × C is a subset of B × D
Here,
A x C = {(1, 5), (1, 6), (2, 5), (2, 6)}
B x D = {(1, 5), (1, 6), (1, 7), (1, 8), (2, 5), (2, 6), (2, 7), (2, 8), (3, 5), (3, 6), (3, 7), (3, 8), (4, 5), (4, 6), (4, 7), (4, 8)}
Since, all the elements of set A x C are the elements of set B x D
Thus, A x C is a subset of B × D
Hence, verified.
问题8.令A = {1，2}和B = {3，4}。写A×B。A×B有多少个子集?列出他们。
解决方案：
Given, A= {1, 2} and B = {3, 4}
So, A × B = {(1, 3), (1, 4), (2, 3), (2, 4)}
Number of elements in A × B = n(A × B) = 4
We know that,
For a set S with n(S) = m, number of subsets of S is given by n[P(S)] = 2m.
Thus, the set A × B has 24 = 16 subsets.
These subsets are: ∅, {(1, 3)}, {(1, 4)}, {(2, 3)}, {(2, 4)}, { (1, 3), (1, 4) }, { (1, 3), (2, 3) }, { (1, 3), (2, 4) }, {(1, 4), (2, 3)}, { (1, 4), (2, 4) }, { (2, 3), (2, 4) }, {(1, 3), (1, 4), (2, 3) }, { (1, 3), (1, 4), (2, 4) }, { (1, 3), (2, 3), (2, 4) }, { (1, 4), (2, 3), (2, 4) }, { (1, 3), (1, 4), (2, 3), (2, 4)}
问题9.令A和B为两组，使得n(A)= 3和n(B)=2。如果(x，1)，(y，2)，(z，1)在A×B中，找到A和B，其中x，y和z是不同的元素。
解决方案：
Given,
n(A) = 3 and n(B) = 2; and (x, 1), (y, 2), (z, 1) are in A × B.
We know that,
A is the set of first elements and B is the set of second elements of the ordered pair elements of A x B.
So, the elements of A are x, y, z and the elements of B are 1, 2
As, n(A) = 3 and n(B) = 2, it is clear that set A = {x, y, z} and set B = {1, 2}.
问题10.笛卡尔乘积A×A有9个元素，其中有(–1，0)和(0,1)。找到集合A和A×A的其余元素。
解决方案：
We know that,
If there are p elements in A and q elements in B, then there will be pq elements in A × B
, i.e., if n(A) = p and n(B) = q, then n(A × B) = pq
Given, n(A × A) = 9
So, n(A) × n(A) = 9
Thus, n(A) = 3
Also given that, the ordered pairs (-1, 0) and (0, 1) are two of the nine elements of A × A.
And, we know A × A = {(a, a): a ∈ A}.
So, -1, 0, and 1 should be the elements of A.
As n(A) = 3, clearly A= {-1, 0, 1}.
Hence, the remaining elements of set A × A are as follows: (-1, -1), (-1, 1), (0, -1), (0, 0), (1, -1), (1, 0), and (1, 1)