Subsets and universal set
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Subsets and universal set

In this video, we will learn about subsets.
Let A be set of all x such that x is a student of your school.
And B be set of all x such that x is a student of your class.
As, every student of your class is also student of your school. So, every element of B is
also an element of A; so, we say that B is a subset of A.
The fact that B is subset of A is expressed in symbols as B elongated C A.
The symbol elongated C stands for ‘is a subset of’ or ‘is contained in’.
A set S is said to be a subset of a set T if every element of S is also an element of
T. In other words, S ? T if whenever element
a belongs to S, then element a belongs to T.
The symbol right arrow means implies. Using this symbol, we can write the definiton of
subset as follows: We read the statement as “S is a subset
of T if a is an element of S implies that a is also an element of T”.
If S is not a subset of T, we write S crossed elongated C T.
It follows from the above definition that every set S is a subset of itself, i.e., S
is a subset of S. Since the empty set phi has no elements, we
agree to say that phi is a subset of every set.
We now consider some examples: The set of rational numbers is represented
by Q. The set of real numbers is represented by
R. As, every rational number is a real number
so Q is a subset of R. If C is the set of all divisors of 56.
And D be the set of all prime divisors of 56.
Then of course D is the subset of C. Let E be the set of vowels of English alphabet
then E is a set of letters a, e, i, o, and u and let F be a set of letters a, b, c, and
d. As, it can be observed clearly E is not a
subset of F, also F is not a subset of E. Let’s look into subsets of set of real numbers
Set of real numbers are represented by R. R cannot be expressed in roster form as elements
of R are infinite and do not follow any pattern. The set of natural numbers is denoted by N.
N={1, 2, 3, 4, 5, . . .} Here is the set of integers denoted by Z.
Z={. . ., –3, –2, –1, 0, 1, 2, 3, . . .}
Here the three dots denotes the pattern will be followed throughout indefinitely.
The set of rational numbers is denoted by Q. Q={ x : x=p/q, p and q belong to set
of integers and q is not equal to 0} We can clearly see that the set of natural
number N is a subset of set of integers Z. AS every natural number is an integer. As
per the definition of the rational number the rational number is of form p upon q. If
q=1 then we can generate every integer. So, set of integers Z is a subset of rational
number Q. As real number consists of all rational numbers and irrational numbers. So, Q is subset
of R. Let’s learn about universal set.
Universal set contains all elements and all sets of interest are subsets of this set.
The universal set is usually denoted by capital letter U.
For example, for the set of all integers, the universal set can be the set of rational
numbers or, for that matter, the set R of real numbers.
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