11类NCERT解决方案–第1章集–练习1.4

问题1.找到以下各对集合的并集：

(i)X = {1、3、5} Y = {1、2、3}

(ii)A = {a，e，i，o，u} B = {a，b，c}

(iii)A = {x：x是自然数，是3的倍数}

B = {x：x是小于6的自然数}

(iv)A = {x：x是自然数，1 < x≤6}

B = {x：x是自然数，6

(v)A = {1,2,3}，B =Φ

解决方案：

(i) X = {1, 3, 5} Y = {1, 2, 3}

So, the union of the pairs of set can be written as

X ∪ Y= {1, 2, 3, 5}

(ii) A = {a, e, i, o, u} B = {a, b, c}

So, the union of the pairs of set can be written as

A∪ B = {a, b, c, e, i, o, u}

(iii) A = {x: x is a natural number and multiple of 3} = {3, 6, 9 …}

B = {x: x is a natural number less than 6} = {1, 2, 3, 4, 5, 6}

So, the union of the pairs of set can be written as

A ∪ B = {1, 2, 4, 5, 3, 6, 9, 12 …}

Hence, A ∪ B = {x: x = 1, 2, 4, 5 or a multiple of 3}

(iv) A = {x: x is a natural number and 1 < x ≤ 6} = {2, 3, 4, 5, 6}

B = {x: x is a natural number and 6 < x < 10} = {7, 8, 9}

So, the union of the pairs of set can be written as

A∪ B = {2, 3, 4, 5, 6, 7, 8, 9}

Hence, A∪ B = {x: x ∈ N and 1 < x < 10}

(v) A = {1, 2, 3}, B = Φ

So, the union of the pairs of set can be written as

A∪ B = {1, 2, 3}

问题2。令A = {a，b}，B = {a，b，c}。是A⊂B吗?什么是A∪B?

解决方案：

It is given that

A = {a, b} and B = {a, b, c}

Yes, A ⊂ B

So, the union of the pairs of set can be written as

A∪ B = {a, b, c} = B

问题3.如果A和B是两个集合，使得A⊂B，那么A∪B是什么?

解决方案：

If A and B are two sets such that A ⊂ B, then A ∪ B = B.

问题4.如果A = {1、2、3、4}，B = {3、4、5、6}，C = {5、6、7、8}和D = {7、8、9、10 };找

(i)A∪B

(ii)A∪C

(iii)B∪C

(iv)B∪D

(v)A∪B∪C

(vi)A∪B∪D

(vii)B∪C∪D

解决方案：

It is given that

A = {1, 2, 3, 4], B = {3, 4, 5, 6}, C = {5, 6, 7, 8} and D = {7, 8, 9, 10}

(i) A ∪ B = {1, 2, 3, 4, 5, 6}

(ii) A ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}

(iii) B ∪ C = {3, 4, 5, 6, 7, 8}

(iv) B ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}

(v) A ∪ B ∪ C = {1, 2, 3, 4, 5, 6, 7, 8}

(vi) A ∪ B ∪ D = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}

(vii) B ∪ C ∪ D = {3, 4, 5, 6, 7, 8, 9, 10}

问题5.找到每对集合的交集：

(i)X = {1、3、5} Y = {1、2、3}

(ii)A = {a，e，i，o，u} B = {a，b，c}

(iii)A = {x：x是自然数，是3的倍数}

B = {x：x是小于6的自然数}

(iv)A = {x：x是自然数，1

B = {x：x是自然数，6

(v)A = {1,2,3}，B =Φ

解决方案：

(i) X = {1, 3, 5}, Y = {1, 2, 3}

So, the intersection of the given set can be written as

X ∩ Y = {1, 3}

(ii) A = {a, e, i, o, u}, B = {a, b, c}

So, the intersection of the given set can be written as

A ∩ B = {a}

(iii) A = {x: x is a natural number and multiple of 3} = (3, 6, 9 …}

B = {x: x is a natural number less than 6} = {1, 2, 3, 4, 5}

So, the intersection of the given set can be written as

A ∩ B = {3}

(iv) A = {x: x is a natural number and 1 < x ≤ 6} = {2, 3, 4, 5, 6}

B = {x: x is a natural number and 6 < x < 10} = {7, 8, 9}

So, the intersection of the given set can be written as

A ∩ B = Φ

(v) A = {1, 2, 3}, B = Φ

So, the intersection of the given set can be written as

A ∩ B = Φ

问题6。如果A = {3,5,7,9,11}，B = {7,9,11,13}，C = {11,13,15}，D = {15,17}；找

(i)A∩B

(ii)B∩C

(iii)A∩C∩D

(iv)A∩C

(v)B∩D

(vi)A∩(B∪C)

(vii)A∩D

(viii)A∩(B∪D)

(ix)(A∩B)∩(B∪C)

[x)(A∪D)∩(B∪C)

解决方案：

(i) A ∩ B = {7, 9, 11}

(ii) B ∩ C = {11, 13}

(iii) A ∩ C ∩ D = {A ∩ C} ∩ D

= {11} ∩ {15, 17}

= Φ

(iv) A ∩ C = {11}

(v) B ∩ D = Φ

(vi) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C)

= {7, 9, 11} ∪ {11}

= {7, 9, 11}

(vii) A ∩ D = Φ

(viii) A ∩ (B ∪ D) = (A ∩ B) ∪ (A ∩ D)

= {7, 9, 11} ∪ Φ  

= {7, 9, 11}

(ix) (A ∩ B) ∩ (B ∪ C) = {7, 9, 11} ∩ {7, 9, 11, 13, 15}

= {7, 9, 11}

(x) (A ∪ D) ∩ (B ∪ C) = {3, 5, 7, 9, 11, 15, 17) ∩ {7, 9, 11, 13, 15}

= {7, 9, 11, 15}

问题7。如果A = {x：x是一个自然数}，B = {x：x是一个偶数自然数}，C = {x：x是一个奇数自然数}，D = {x：x是一个自然数}素数}，找到

(i)A∩B

(ii)A∩C

(iii)A∩D

(iv)B∩C

(v)B∩D

(vi)C∩D

解决方案：

It can be written as

A = {x: x is a natural number} = {1, 2, 3, 4, 5 …}

B ={x: x is an even natural number} = {2, 4, 6, 8 …}

C = {x: x is an odd natural number} = {1, 3, 5, 7, 9 …}

D = {x: x is a prime number} = {2, 3, 5, 7 …}

(i) A ∩B = {x: x is a even natural number} = B

(ii) A ∩ C = {x: x is an odd natural number} = C

(iii) A ∩ D = {x: x is a prime number} = D

(iv) B ∩ C = Φ

(v) B ∩ D = {2}

(vi) C ∩ D = {x: x is odd prime number}

问题8.以下哪些对集合是不相交的

(i){1,2,3,4}和{x：x是自然数，且4≤x≤6 }

(ii){a，e，i，o，u}和{c，d，e，f}

(iii){x：x是偶数整数}，{x：x是奇数整数}

解决方案：

(i) {1, 2, 3, 4}

{x: x is a natural number and 4 ≤ x ≤ 6} = {4, 5, 6}

So, we get

{1, 2, 3, 4} ∩ {4, 5, 6} = {4}

Hence, this pair of sets is not disjoint.

(ii) {a, e, i, o, u} ∩ (c, d, e, f} = {e}

Hence, {a, e, i, o, u} and (c, d, e, f} are not disjoint.

(iii) {x: x is an even integer} ∩ {x: x is an odd integer} = Φ

Hence, this pair of sets is disjoint.

问题9.如果A = {3，6，9，12，15，18，21}，B = {4，8，12，16，20}，C = {2，4，6，8，8，10，12 ，14，14，}，D = {5，10，15，20};找

(i)A – B

(ii)A – C

(iii)A – D

(iv)B – A

(v)C – A

(vi)D – A

(vii)B – C

(viii)B – D

(ix)C – B

(x)D – B

(xi)C – D

(xii)D – C

解决方案：

(i) A – B = {3, 6, 9, 15, 18, 21}

(ii) A – C = {3, 9, 15, 18, 21}

(iii) A – D = {3, 6, 9, 12, 18, 21}

(iv) B – A = {4, 8, 16, 20}

(v) C – A = {2, 4, 8, 10, 14, 16}

(vi) D – A = {5, 10, 20}

(vii) B – C = {20}

(viii) B – D = {4, 8, 12, 16}

(ix) C – B = {2, 6, 10, 14}

(x) D – B = {5, 10, 15}

(xi) C – D = {2, 4, 6, 8, 12, 14, 16}

(xii) D – C = {5, 15, 20}

问题10。如果X = {a，b，c，d}并且Y = {f，b，d，g}，则找到

(i)X – Y

(ii)Y – X

(iii)X∩Y

解决方案：

(i) X – Y = {a, c}

(ii) Y – X = {f, g}

(iii) X ∩ Y = {b, d}

问题11.如果R是实数集合，而Q是有理数集合，那么R – Q是什么?

解决方案：

We know that

R – Set of real numbers

Q – Set of rational numbers

Hence, R – Q is a set of irrational numbers.

问题12。说明以下每个陈述是对还是错。证明你的答案。

(i){2，3，4，5}和{3，6}是不交集。

(ii){a，e，i，o，u}和{a，b，c，d}是不相交的集合。

(iii){2，6，10，14}和{3，7，11，15}是不交集。

(iv){2，6，10}和{3，7，11}是不交集。

解决方案：

(i) False

If 3 ∈ {2, 3, 4, 5}, 3 ∈ {3, 6}

So, we get {2, 3, 4, 5} ∩ {3, 6} = {3}

(ii) False

If a ∈ {a, e, i, o, u}, a ∈ {a, b, c, d}

So, we get {a, e, i, o, u} ∩ {a, b, c, d} = {a}

(iii) True

Here {2, 6, 10, 14} ∩ {3, 7, 11, 15} = Φ

(iv) True

Here {2, 6, 10} ∩ {3, 7, 11} = Φ