Basic Level Examination
What is a Set?
In mathematics, a set is a well-defined collection of distinct objects. These objects can be anything: numbers, symbols, elements, or even other sets. The defining characteristic of a set is that it does not contain any duplicate elements, and the order of the elements is not considered significant.
Sets are typically denoted by curly braces {}, with the elements listed inside separated by commas. For example, the set of even numbers less than 10 can be represented as {2, 4, 6, 8}.
Sets are fundamental in various mathematical concepts and disciplines, such as set theory, algebra, and calculus. They serve as building blocks for defining more complex mathematical structures and operations.
What is a Well-defined Object?
A “well-defined” object, particularly in mathematics, refers to something that has clear, unambiguous properties or characteristics that allow it to be uniquely identified or described without any ambiguity. When defining an object, it’s essential to ensure that the definition is precise and does not lead to any contradictions or inconsistencies.
For example, consider the following set: {x : x is an even number}. This set is well-defined because the property “x is an even number” clearly specifies which elements belong to the set (i.e., 2, 4, 6, etc.).
In contrast, an object or definition that is not well-defined may be ambiguous, vague, or contradictory, making it difficult or impossible to determine its properties or characteristics conclusively.
What are the types of set?
Sets can be classified into various types based on different criteria. Some common types of sets include:
- Finite Set: A set that contains a specific number of elements. For example, {1, 2, 3, 4} is a finite set because it has four elements.
- Infinite Set: A set that contains infinite numbers of elements. For example, the set of all natural numbers {1, 2, 3, …} is an infinite set.
- Empty Set or Null Set: A set that contains no elements. It is denoted by the symbol { } or ∅. For example,{ } or ∅ represents the empty set.
- Singleton Set: A set that contains only one element. For example, {5} is a singleton set because it contains only the element 5.
- Equal Set: Two sets are equal if they have exactly the same elements, regardless of their order of listing. For example, {1, 2, 3} and {2, 3, 1} are equal sets.
Super Set and Subset
In the given figure,
Red circle represents Set A and Black Circle represents Set B.
Set A contains a, e as well as the whole set B.
Hence,
A={a, b, c, d, e}and B={b, c, d}
Set A is said to be subset of set B if at least one element of set A is also the member of set B.
It is denoted by either ⊂ or ⊆ symbol.
In above condition where A={a, b, c, d, e}and B={b, c, d},
B is subset of A and it is written as B⊂A and read as B is subset of A.
[NOTE: Between two Sets, if one set is subset than other set is called super set.For example,
B is subset of A and it is written as B⊂A and read as B is subset of A.
i.e. Set A is Superset of Set B. It is written as A⊃B and read as A is superset of B.]
There are two types of subsets.
- Proper Subsets (⊂)
- Improper Subsets (⊆)
Proper Subset
- Proper Subsets:
Set A is said to be proper subset of set B if at least one member (but not all members) of set A is also the member of set B.
For example,
If P={1, 2, 3. 4. 5. 6. 7} and Q={2, 4, 6} then,
Set Q is proper subset of Set P. Written as Q ⊂ P.
In the figure, Set Q lies inside the Set P
- Improper Subsets (⊆)
Set A is said to be improper subset of set B if all members of set A is also the members of set B.
For example,
If P={1, 2, 3. 4. 5. 6. 7} and Q={1, 2, 3. 4. 5. 6. 7} then,
Set P is improper subset of Set Q. Written as P⊆Q.
[NOTE: In the figure, P and Q will have same circle or oval.]Relationship of set
- Equal Set or Equality: Two sets are considered equal if they contain exactly the same elements. In other words, if every element of set A is also an element of set B, and vice versa, then A = B. For example, If A = {first five English alphabets} and and B = {a, b, c, d, e} then set A and B are said to be an equal set. So, Set A = Set B
- Equivalent Set: Two sets are said to be equivalent Set if they contain equal number of elements. It is denoted by
For example,
If A={a, b, c, d, e} and B={1, 2, 3, 4, 5}, Set A and Set B have 5 elements on both of them.
So, Set A is equivalent to Set B and Written as A~B.
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5. Disjoint Sets: Two sets are said to be the disjoint set if they do not have any common elements.
In other words, their intersection is the empty set (∅). For example, If A = {1, 2} and B = {3, 4}, then A and B are disjoint sets.
In the given figure,
A={Nepal, Pakistan, Bhutan, Srilanka}
B={India, Afghanistan, Maldives, Bangladesh}
They do not have any common elements. So, Set A and Set B are said to be disjoint set.
6. Overlapping Set: Two sets are said to be overlapping set if they have at least one common element. In the given figure, set A={a, e, I, o, u}
Set B={a, b, c, d, e}
They have “a” in common (i.e. “a” is on the both sets A and B). So Set A and Set B are said to be Overlapping Set.
