Equal Set: Two sets, A and B, are said to be equal if and only if A is a subset of B and B is a subset of A. To indicate that two sets, A and B, are equal, we use the symbol A = B.
This means that sets A and B contain exactly the same elements. A ≠ B means that A and B are not equal sets.
Example:
Let A be the set of even natural numbers and B be the set of natural numbers which are multiples of 2. Clearly, A ⊂ B and B ⊂ A. Therefore, since A and B contain exactly the same elements, A = B.
Proper Subset :
If the number of elements of any subset formed from a set is less than the given set, it is called the proper subset.
A={a,b} B={a}C={b}. so B,C is proper subset
Universal Set :
All sets related to the discussions are subset of a definite set. Such as, A={x,y} is a subset of B={x,y,z}. Here, set B is called the universal set in with respect to the set A
intersection: Let A and B be subsets of a set X. The intersection of A and B is the set of all elements in X common to both A and B.
Notation: “A ⋂ B” denotes “A intersection B” or the intersection of sets A and B.
Thus, A ⋂ B = {x ∈ X | x ∈ A and x ∈ B}, or A ⋂ B = {x | x ∈ A Ù x ∈ B}.
Examples:
Let A = {2,4,5} and B = {1,4,6,8} Then, A ⋂ B = {4}.
Note: A set that has only one element, such as {4}, is sometimes called a singleton set.
Let A = {2,4,5} and B = {1,3}. Then A ⋂ B = ∅.
* If, A and B are two sets such that A ⋂ B is the empty set, we say that A and B are disjoint.
*Given sets A and B. x ∈ A ⋂ B if and only if x ∈ A and x ∈ B.
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