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$$A=\left\{\text{x: x is a vowel in English alphabet}\right\}$$
$$B=\left\{\text{x: x is an integer and}\phantom{\rule{2pt}{0ex}}0<x<10\right\}$$
The roaster equivalents of two sets are :
$$A=\{a,e,i,o,u\}$$
$$B=\left\{\mathrm{1,2,3,4,5,6,7,8,9}\right\}$$
Can we write set “B” as the one which comprises single digit natural number? Yes. Thus, we can see that there are indeed different ways to define and identify members and hence the flexibility in defining collection.
We should be careful in using words like “and” and “or” in writing qualification for the set. Consider the example here :
$$C=\{\text{x:}x\in Z\phantom{\rule{2pt}{0ex}}\text{and}\phantom{\rule{2pt}{0ex}}2<x<4\}$$
Both conditional qualifications are used to determine the collection. The elements of the set as defined above are integers. Thus, the only member of the set is “3”.
Now, let us consider an example, which involves “or” in the qualification,
$$C=\{\text{x:}\phantom{\rule{2pt}{0ex}}x\in A\phantom{\rule{2pt}{0ex}}\text{or}\phantom{\rule{2pt}{0ex}}x\in B\}$$
The member of this set can be elements belonging to either of two sets "A" and "B". The set consists of elements (i) belonging exclusively to set "A", (ii) elements belonging exclusively to set "B" and (iii) elements common to sets "A" and "B".
Problem 1 : A set in roaster form is given as :
$$A=\{\frac{{5}^{2}}{6},\frac{{6}^{2}}{7},\frac{{7}^{2}}{8}\}$$
Write the set in “set builder form”.
Solution : We see here that we are dealing with natural numbers. The numerators are square of natural numbers in sequence. The number in denominator is one more than numerator for each member. We can denote natural number by “n”. Clearly, if numerator is “ ${n}^{2}$ ”, then denominator is “n+1”. Therefore, the expression that represent a member of the set is :
$$x=\frac{{n}^{2}}{n+1}$$
However, this set is not an infinite set. It has exactly three members. Therefore, we need to specify “n” so that only members of the set are exclusively denoted by the above expression. We see here that “n” is greater than 4, but “n” is less than 8. For representing three elements of the set,
$$5\le n\le 7$$
We can write the set, now, in the builder form as :
$$A=\{x:\phantom{\rule{1em}{0ex}}x=\frac{{n}^{2}}{n+1},\text{where "n" is a natural number and}\phantom{\rule{1em}{0ex}}5\le n\le 7\}$$
In set builder form, the sequence within the range is implied. It means that we start with the first valid natural number and proceed sequentially till the last valid natural number.
Few key number sets are used regularly in mathematical context. As we use these sets often, it is convenient to have predefined symbols :
We put a superscript “+”, if we want to specify membership of only positive numbers, where appropriate. " ${Z}^{+}$ ", for example, means set of positive integers.
An empty set has no member or object. It is denoted by symbol “φ” and is represented by a pair of braces without any member or object.
$$\phi =\left\{\right\}$$
The empty set is also called “null” or “void” set. For example, consider a definition : “the set of integer between 1 and 2”. There is no integer within this range. Hence, the set corresponding to this definition is an empty set. Consider another example :
$$B=\{x:\phantom{\rule{1em}{0ex}}{x}^{2}=4\phantom{\rule{1em}{0ex}}\text{and x is odd}\}$$
An odd integer squared can not be even. Hence, set “B” also does not have any element in it.
There is a bit of paradox here. If the definition does not yield an element, then the set is not well defined. We may be tempted to say that empty set is not a set in the first place. However, there is a practical reason to have an empty set. It enables mathematical operations. We shall find many examples as we study operations on sets.
The members of two equal sets are exactly same. There is nothing more to it. However, we need to know two special aspects of this equality. We mentioned about repetition of elements in a set. We observed that repetition of elements does not change the set. Consider example here :
$$A=\left\{\mathrm{1,5,5,8,7}\right\}=\left\{\mathrm{1,5,}\mathrm{8,7}\right\}$$
Another point is that sequence does not change the set. Therefore,
$$A=\left\{\mathrm{1,5,8,7}\right\}=\left\{\mathrm{5,7,8,1}\right\}$$
In the nutshell, when we have to compare two sets we look for distinct elements only. If they are same, then two sets in question are equal.
Cardinality is the numbers of elements in a set. It is denoted by modulus of set like |A|.
The cardinality of an empty set is zero. The cardinality of a finite set is some positive integers. The cardinality of a number system like integers is infinity. Curiously, the cardinality of some infinite set can be compared. For example, the cardinality of natural numbers is less than that of integers. However, we can not make such deduction for the most case of infinite sets.
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