问题1(i)。如果(a / 3 + 1，b – 2/3)=(5/3，1/3)，则找到a和b的值。
解决方案：
According to the definition of equality of ordered pairs
(a/3 + 1, b – 2/3) = (5/3, 1/3)
⇒ a/3 + 1 = 5/3 and b – 2/3 =1/3
⇒ a/3 = (5 – 3)/3 and b = (1/3 + 2/3)
⇒ a/3 = 2/3 and b = 3/3
⇒ a = 2 and b = 1
问题1(ii)。如果(x + 1，1)=(3，y – 2)，则求出x和y的值。
解决方案：
According to the definition of equality of ordered pairs
(x + 1, 1) = (3, y – 2)
⇒ x + 1 = 3 and 1 = y – 2
⇒ x = 3 – 1 and 1 + 2 = y
⇒ x = 2 and 3 = y
⇒ x = 2 and y = 3
问题2.如果有序对(x，-1)和(5，y)属于集合{(a，b)：b = 2a – 3}，则求x和y的值。
解决方案：
Given:
(x, -1) ∈ {(a, b) : b = 2a – 3}
and, (5, y) ∈ {(a, b) : b = 2a – 3}
⇒ -1 = 2 × x – 3 and y = 2 × 5 – 3
⇒ -1 = 2x – 3 and y = 10 – 3
⇒ 3 – 1 = 2x and y = 7
⇒ 2 = 2x and y = 7
⇒ x = 1 and y = 7
问题3。如果a∈{-1，2，3，4，5}和b∈{0，3，6}，则写出所有有序对(a，b)的集合，使得a + b = 5。
解决方案：
Given: a ∈ {- 1, 2, 3, 4, 5} and b ∈ {0, 3, 6},
Now, we have to find the ordered pair (a, b) such that a + b = 5
So, the ordered pair (a, b) such that a + b = 5 are as follows
(a, b) ∈ {(- 1, 6), (2, 3), (5, 0)}
问题4.如果a∈{2，4，6，9}和b∈{4，6，18，27}，则形成所有有序对(a，b)的集合，使得a除以b并且a
解决方案：
Given: a ∈ {2, 4, 6, 9} and b ∈ {4, 6, 18, 27}
Here,
2 divides 4, 6, 18 and is also less than all of them
4 divides 4 and is also less than none of them
6 divides 6, 18 and is less than 18 only
9 divides 18, 27 and is less than all of them
∴ Ordered pairs (a, b) are (2, 4), (2, 6), (2, 18), (6, 18), (9, 18) and (9, 27)
问题5.如果A = {1，2}并且B = {1，3}，则找到A x B和B xA。
解决方案：
Given: A = {1, 2} and B = {1, 3}
Now we have to find A x B, and B x A
A × B = {1, 2} × {1, 3}
= {(1, 1), (1, 3), (2, 1), (2, 3)}
B × A = {1, 3} × {1, 2}
= {(1, 1), (1, 2), (3, 1), (3, 2)}
问题6.令A = {1,2,3}，B = {3，4}。找到A x B并以图形方式显示
解决方案：
Given: A = {1, 2, 3} and B = {3, 4}
Now we have to find A x B
A x B = {1, 2, 3} × {3, 4}
= {(1, 3), (1, 4), (2, 3), (2, 4), (3, 3), (3, 4)}
To draw A x B graphically follow the following steps:
Step 1: Draw horizontal and vertical axis.
Step 2: The horizontal axis represents set A and the vertical axis represents set B.
Step 3: Now, draw dotted lines perpendicular to horizontal and vertical axes through the elements of set A and B
Step 4: Point of intersection of these perpendicular represents A × B
问题7。如果A = {1,2,3}，B = {2，4}，则A x B，B x A，A x A，B x B和(A x B)what(B x一种)?
解决方案：
Given:
A = {1, 2, 3} and B = {2, 4}
Now we have to find A × B, B × A, A × A, and (A × B) ∩ (B × A)
A × B = {1, 2, 3} × {2, 4}
= {(1, 2), (1, 4), (2, 2), (2, 4), (3, 2), (3, 4)}
B × A = {2, 4} × {1, 2, 3}
= {(2, 1), (2, 2), (2, 3), (4, 1), (4, 2), (4, 3)}
A × A = {1, 2, 3} × {1, 2, 3}
= {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
B × B = {2, 4} × {2, 4}
= {(2, 2), (2, 4), (4, 2), (4, 4)}
Intersection of two sets represents common elements of both the sets
So,
(A × B) ∩ (B × A) = {(2, 2)}
问题8.如果A和B的两个集合具有3个共同点。如果n(A)= 5，n(B)= 4，求n(A×B)和n [(A×B)∩(B×A)]
解决方案：
Given: n(A) = 5 and n(B) = 4
We know that if A and B are two finite sets, then n(A × B) = n(A) × n(B)
Therefore,
n(A × B) = 5 × 4 = 20
Now,
n[(A × B) ∩ (B × A)] = 3 × 3 = 9 -(∵ A and B have 3 common elements)
问题9.让A和B两组。证明如果集合A和B具有共同的元素，则集合A×B和B×A具有共同的元素。
解决方案：
Let us considered(a, b) be an arbitrary elements of (A × B) ∩ (B × A). Then,
(a, b) ∈ (A × B) ∩ (B × A)
= (a, b) ∈ A × B and (a, b) ∈ B × A
= (a ∈ A and b ∈ B) and (a ∈ B and b ∈ A)
= (a ∈ A and a ∈ B) and (b ∈ A and b ∈ B)
= a ∈ A ∩ B and b ∈ A ∩ B
Hence, the sets A × B and B × A have an element in
common have an element in common.
问题10.设A和B两个集合，使得n(A)= 3和n(B)=2。如果(x，1)，(y，2)，(z，1)在A×B中，则找到A和B，其中x，y，z是不同的元素
解决方案：
Since (x, 1), (y, 2), (z, 1) are elements of A × B. Therefore, x, y, z ∈ A and 1, 2 ∈ B
Given: n(A) = 3 and n(B) = 2
Therefore, x, y, z ∈ A and n(A) = 3
⇒ A = (x, y, z)
1, 2 ∈ B and n(B) = 2
⇒ B = (1, 2)
问题11。设A = {1,2,3,4}，R = {(a，b)：a∈A，b∈A，a除以b}。显式地写R。
解决方案：
Given: A = (1, 2, 3, 4) and, R = {(a, b) : a ∈ A, b ∈ A, a divides b}
Now, a/b stands for ‘a divides b’.
So, for the elements of the given sets, we find that 1/1, 1/2, 1/3, 1/4, 2/2, 3/3 and 4/4
Therefore,
R = {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}
问题12。如果A = {-1，1}，则求A×A×A
解决方案：
Given: A = {-1, 1}
So, A × A = {-1, 1} × {-1, 1}
= {(-1, -1), (-1, 1), (1, -1), (1, 1)}
Therefore, 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, -1), (1, -1, 1), (1, 1, -1), (1, 1, 1)}
问题13.陈述以下每个陈述是对还是错。如果语句为假，则正确地重写给定的语句：
(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∈B和y∈A。
(iii)如果A = {1，2}并且B = {3，4}，则A∩(B∩)=∅
解决方案：
(i) False,
If P = {m, n} and Q = {n, m},
Then,
P × Q = {(m, n), (m, m), (n, n), (n, m)}
(ii) False,
If A and B are non-empty sets, then AB is a non-empty set of ordered pairs (x, y) such that x ∈ A and y ∈ B
(iii) True
问题14.如果A = {1，2}形成集合A×A×A
解决方案：
Given: A = {1, 2}
So, A × A = {1, 2} × {1, 2}
= {(1, 1), (1, 2), (2, 1), (2, 2)}
Therefore,
A × A × A = {1, 2} × {(1, 1), (1, 2), (2, 1), (2, 2)}
= {(1, 1, 1), (1, 1, 2), (1, 2, 1), (1, 2, 2), (2, 1, 1), (2, 1, 2), (2, 2, 1), (2, 2, 2)}
问题15(i)。如果A = {1,2,4}且B = {1,2,3}，则以图形方式表示A×B
解决方案：
Given: A = {1, 2, 4} and B = {1, 2, 3}
So, A × B = {1, 2, 4} × {1, 2, 3}
= {(1, 1), (1, 2), (1, 3), (2, 1), (2, 2), (2, 3), (4, 1), (4, 2), (4, 3)}
Hence, we represent A on the horizontal line and B on vertical line.
So, the graphical representation of A × B is as shown below:
问题15(ii)。如果A = {1,2,4}且B = {1,2,3}，则用图形表示B×A。
解决方案：
Given: A = {1, 2, 4} and B = {1, 2, 3}
So, B × A = {1, 2, 3} × {1, 2, 4}
= {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (2, 4), (3, 1), (3, 2), (3, 4)}
Hence, we represent B on the horizontal line and A on vertical line.
So, the graphical representation of B × A:
问题15(iii)。如果A = {1,2,4}且B = {1,2,3}，则用图形表示A×A。
解决方案：
Given: A = {1, 2, 4}, B = {1, 2, 3}
So, A × A = {1, 2, 4} × {1, 2, 4}
= {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (2, 4), (4, 1), (4, 2), (4, 4)}
Graphical representation of A × A:
问题15(iv)。如果A = {1，2，4}并且B = {1,2,3}，则用图形表示B×B。
解决方案：
Given: A = {1, 2, 4}, B = {1, 2, 3}
So, B × B = {1, 2, 3} × {1, 2, 3}
= {(1, 1), (1, 2), (1, 4), (2, 1), (2, 2), (2, 3), (3, 1), (3, 2), (3, 3)}
Graphical representation of B × B:
