SETS
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De Morgan’s First Law The complement of the union of two sets is equal to the intersection of their complements i.e. (A ∪ B )' = A' ∩ B' De Morgan’s Second Law The complement of the intersection of two sets is equal to the union of their complements i.e. (A ∩ B )’ = A’ ∪ B’ READ MORE...

A cartesian product between two sets is defined as the set consisting of all possible ordered pairs that can be formed by taking one element from each of the sets at a given time. If A and B are two sets such that a ∈ A and b ∈ B, then the cartesian product between A and B is denoted as A × B and is evaluated as { (a,b) } where a ∈ A and b ∈ B. Let A = { 1,2 } and B = { x,y } The cartesian product of A and B denoted as A × B = { (1,x) , (1,y), (2,x), (2,y) } The cartesian product of B and A, denoted as […] READ MORE...

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...