Set Concepts (Some Simple Divisions by Vedic Method)
ICSE Class 7 · Mathematics · Chapter 13
What this chapter covers
- Sets: a quick revision
- Cardinal number of a set
- Equal sets
- Equivalent sets
- Universal set
- Subsets
- Simple division by the Vedic method
Key definitions
Equal sets are two sets that have exactly the same elements.
Equivalent sets are two finite sets that have the same cardinal number.
Vedic mathematics is a collection of short calculation methods, each based on a brief rule called a sutra, that let you do many sums mentally or in very few written steps.
Key points
- A set is a well-defined collection of distinct objects; its members are called elements.
- A set can be written in roster form (listing) or set-builder form (stating a rule).
- Sets can be finite, infinite, empty or singleton.
- The cardinal number n(A) counts the distinct elements of a finite set; n(φ) = 0.
- Equal sets have exactly the same elements; equivalent sets have the same cardinal number.
- All equal sets are equivalent, but equivalent sets need not be equal.
- A universal set xi holds every element under discussion and is drawn as a rectangle in a Venn diagram.
- A subseteq B when every element of A is in B; every set is a subset of itself, and φ is a subset of every set.
- A proper subset leaves out at least one element of the larger set.
- For the Vedic method of division, find the base and the shortfall, split the dividend with a bar, and build the quotient by multiplying by the shortfall and adding.
- A remainder that is not less than the divisor must be divided again, with the extra added to the quotient.
- Always check: dividend = divisor × quotient + remainder.
Questions answered
Is "the clever students of Class VII" a set? Give a reason.
No. "Clever" is not well-defined, so different people would choose different students.
Write \x : x is a prime number, x < 15\ in roster form and find its cardinal number.
\2, 3, 5, 7, 11, 13\; the cardinal number is 6.
Find n(A) if A is the set of letters in the word COMMITTEE.
A = \c, o, m, i, t, e\, so n(A) = 6.
Are \x : x is a factor of 9\ and \1, 3, 9\ equal sets?
Yes. The factors of 9 are exactly 1, 3 and 9.
Give an example of two sets that are equivalent but not equal.
One example is \a, b, c\ and \7, 8, 9\: both have 3 elements, but the elements differ.
If xi is the set of whole numbers from 0 to 10, list the set of odd numbers in xi.
\1, 3, 5, 7, 9\.
Write all the subsets of \1, 2\ and name the proper subsets.
φ, \1\, \2\ and \1, 2\. The proper subsets are φ, \1\ and \2\.
Which statement is correct: 5 in \5, 6\ or 5 subseteq \5, 6\?
5 in \5, 6\. The number 5 is an element; only a set such as \5\ can be a subset.