MATHEMATICS
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An ordered pair is a 2-tuple formed by taking two elements (generally numbers but can be alphabets, characters, words, or symbols). The general form of representation is (a, b) where a and b represent two distinct objects. The important thing with ordered pairs is that the ordering of the participating elements is important i.e. (a, b) is different from (b, a) unless a=b (a,b) ≠ (b,a) unless a=b Examples of Ordered pairs : (1,2)(a,b)(-172,45.98)(x,3) Remember ☞ (a, b) ≠ (b, a), unless a = b.  If (a1, b2) = (a2, b2) ⇒ a1=a2 & b1=b2 Ordered pairs are widely used in set theory, calculus, relations, and function theories and in the representation of intervals for functions on numbers lines, and axis, for laws and […] READ MORE...

The cartesian product of two sets A and B is defined as a set formed by all the possible ordered pairs of elements from A and B, such that the first element comes from set A and the second element comes from set B. The cartesian product is denoted as A × B. ​Let a and b represent arbitrary elements from set A and set B respectively. So, a ∈ A and b ∈ B. Then A × B = { (a,b) | a ∈ A, b ∈ B }. Remember ☞ ​If A has m elements and B has n elements then their cartesian product will have m × n elements. If n(A) = m and n(B)=n ,  then n(A × B) […] READ MORE...

The intersection of two sets A & B is defined as a set that contains only those members which are common to both A and B. The intersection operation is denoted by the symbol ∩. Remember, for two disjoint sets (sets having no common elements), the intersection results in an empty set (∅). Example ⇒If A = { 1, 2, 3, 4 } and B = { 1, 4, 5, 6 }Then A ∩ B = { 1, 4} Similarly, if A = { 1, 1, 1, 2, 3 } and B = { 1, 1, 2, 2, 4 }Then A ∩ B = { 1, 2 } Similarly, if A = { dog, cat, cow } and B = { peacock, bull […] READ MORE...

The union of two sets A & B is defined as a set that contains all the member elements of A and B. the union operation is denoted by the symbol ∪. One point to remember here is that the union of two or more sets always gives a set with distinct members i.e., in case any of the sets contain any duplicate values or if the result of combining the elements of the set results in duplicate values, the net outcome of the union operation will always result in distinct elements. Example ⇒ If A = { 1, 2, 3, 4 } and B = { 1, 4, 5, 6 } Then A ∪ B = { 1, 2, 3, 4, 5, […] READ MORE...

The complement of a set A (that is a subset of a universal set U) is defined as a set that contains all the member elements and all subsets of U that are not part of the set A. The complement of a set is denoted using the symbol ‘ or c. Hence, A’ = Ac = complement of A Example:- If N is the universal set, and if A = { 1, 2,3 } then A’ = { x ∈ N | x >3 } A = { set of all boys in a class }, then A’ ={ set of all girls in a class } A = { set of all vowels in English language }, then A’ = { set of all consonants […] READ MORE...

Venn diagrams are the pictorial or graphical representation of sets and the various relationships that exist between sets. The representation consists of a rectangular box representing the universal set(U). All sets that are in context are drawn as circles and within the area of the rectangular box. These diagrams were devised by John Venn and hence they are named after him. A typical layout of a Venn diagram is shown below. READ MORE...

When working with sets, a reference superset that contains all the sets and their subsets in context (i.e. the sets which are currently under consideration) is called a universal set. It provides a base set from which all of the sets of interest can be derived or deduced. The universal set concept is needed at times in solving computational problems related to sets and providing certain axioms and in the application of Venn diagrams. When dealing with multiple collections of the same objects, then a set that contains all the possible objects and all sets and subsets formed from these elements is called a superset of every other set or universal set. It is generally denoted by the U. There are no laws […] READ MORE...