集合的表示
想象一个非常杂乱无章的世界,没有分类来单独记忆和分类事物,这样的世界将充满混乱和混乱,这就是为什么人类更喜欢对事物进行分类和分类以便整齐地理解和记住它们的原因.同样的情况也发生在数学中,学习数学涉及处理大量数据,当数据可以分组时,最好将它们分组并分类,因此,集合发挥了作用。
套
集合被定义为定义明确的数据的集合。在数学中,集合是帮助分类和收集属于同一类别的数据的工具,即使集合中使用的元素彼此不同,但它们都相似,因为它们属于一个组。例如,一组不同的户外运动,比如 set A= {Football, Basketball, volleyball, cricket, badminton} 所有提到的运动都是不同的,但它们在一个方面都相似,因为它们属于同一组(户外运动)。
集合用大写字母表示,例如集合A、集合B等,属于集合的元素用小写字母表示,用大括号{}保持,例如集合A= {a, b, c, d},显然a, b, c, d属于集合A,可以写成a∈A,p属于集合A吗?不,因此,它会写成,p∉ A。
集合的表示
集合可以用两种方式表示,一种称为 Roster 形式,另一种称为 Set-Builder 形式,这两种形式可以用来表示相同的数据,只是两种情况下的风格不同。
名册表格
在名册表格中,元素位于 {}⇢ 大括号内。所有元素都在里面提到,并用逗号分隔。名册表格是表示分组数据的最简单方法。例如,表格 5 的集合将是,A= {5, 10, 15, 20, 25, 30, 35.....}。
集合的花名册形式的属性:
- The arrangement in the Roster form does not necessarily to be in the same order every time. For example, A= {a, b, c, d, e} is equal to A= {e, d, a, c, b}.
- The elements are not repeated in the set in Roster form, for example, the word “apple” will be written as, A= {a, p, l, e}
- The Finite sets are represented either with all the elements or if the elements are too much, they are represented as dots in the middle. The infinite sets are represented with dots in the end.
集合生成器表单
在 Set-builder 形式中,元素在表达元素之间关系的语句中显示或表示。 Set-builder 的标准格式,A= {a: statement}。例如,A = {x: x = a 3 , a ∈ N, a < 9}
Set-builder 形式的属性:
- In order to write the set in Set- builder form, the data should follow a certain pattern.
- Colons (:) are necessary in Set-builder form.
- After colon, the statement is to be written.
集合的顺序
Set 的顺序由 Set 中存在的元素数量决定。例如,如果集合中有 10 个元素,则集合的阶数变为 10。对于有限集,集合的阶数是有限的,对于无限集,集合的阶数是无限的。
示例问题
问题1:确定以下哪些被认为是集合,哪些不是。
- 数轴上的所有偶数。
- 九班所有优秀的篮球运动员。
- 这批舞者中表现不佳的人。
- 从 1 到 100 的所有素数。
- 大于 5 且小于 15 的数字。
回答:
Sets are not those bunches or groups where some quality or characteristic comes in the picture. Therefore,
- “All even numbers on the number line” is a set.
- “All the good basketball players from class 9th” is not a Set as “good” is a quality which is involved.
- “The bad performers from the batch of dancers” cannot be a Set since “bad” is a characteristic.
- “All prime numbers from 1 to 100” is a Set.
- “Numbers that are greater than 5 and less than 15” is a Set.
问题 2:在名册表中填写以下信息。
- 所有自然数。
- 大于 6 小于 3 的数字。
- 从 10 到 25 的所有偶数。
回答:
The Roster form for the above information,
- Set A= {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11……}
- Set B= {} ⇢ Null set, since there are no numbers greater than 6 and less than 3.
- Set C= {10, 12, 14, 16, 18, 20, 22, 24}
问题 3:在 Set-Builder 表单中表达给定的信息。
- 大于 10 小于 20 的数字。
- 所有大于 25 的自然数。
- 英文字母中的元音。
回答:
The Set-Builder form for the above information,
- A= {a: a∈ N and 10< a > 20}
- B= {b: b∈ N and b > 25}
- C= {c: c is the vowel of English Alphabet}
问题 4:将以下以 Roster 形式给出的 Set 转换为 Set -Builder 形式。
- A = {b,c,d,f,g,h,j,k,l,m,n,p,q,r,s,t,v,w,x,y,z}
- B= {2, 4, 6, 8, 10}
- C= {5, 7, 9, 11,13, 15, 17, 19}
回答:
The Set- builder form for the above Sets,
- A= {a: a is a consonant of the English Alphabet}
- B= {b: b is an Even number and 2≤ b ≤10}
- C= {c: c is an odd number and 5≤ c ≤ 19}
问题 5:以 Roster 形式和 Set-builder 形式给出以下类型的集合的示例。
- 单数集。
- 有限集。
- 无限集。
解决方案:
The Examples can be taken as per choice since there can be a infinite number of examples for any of the above Sets,
- Singular Set
Roster Form: A= {2}
Set- builder form: A= {a: a∈N and 1 Roster Form: B= {0,1, 2, 3, 4, 5} Set-builder form: B= {b: b is a whole number and b<6} Roster Form: C= {2, 4, 6, 8, 10, 12, 14, 16…..} Set- builder form: C= {c: c is a Natural and Even number}
问题6:给定集合的顺序是什么,
- A= {7, 14, 21, 28, 35}
- B= {a, b, c, d, e, f, e….x, y, z}
- C= {2, 4, 6, 8, 10, 12, 14……}
回答:
The order of the set tells the number of element present in the Set.
- The order of Set A is 5 as it has 5 elements.
- The order of set B is 26 as the English Alphabets have 26 letters.
- The order of set C is infinite as the set has the infinite number of elements.
问题 7:以花名册形式表达给定的集合,
- A = {a: a = n/2, n ∈ N, n < 10}
- B = {b: b = n 2 , n ∈ N, n ≤ 5}
回答:
Representing the above Set-builder sets in Roster form,
- A= {1/2, 1, 3/2, 2, 5/2, 3, 7/2, 4, 9/2}
- B= {1, 4, 9, 16, 25}