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A number not divisible by 2 is called an odd number. Any number whose unit digit(last digit) is either 1,3,5,7 or 9 is an odd number. The set of all odd numbers is represented as Odd numbers = { 2n+1: n ∈  Z } where Z is the set of all integers. When an even number is subtracted from an odd number, it results in an odd number. When two odd numbers are subtracted, it results in an even number. READ MORE...

A number divisible by 2 is called an even number. All numbers whose unit digit(last digit) is either 0,2,4,6 or 8 is an even numbers. The set of all even numbers is represented as follows: Even numbers = { 2n: n ∈  Z } where Z is the set of all integers. Zero is an even number. The addition or subtraction of two even numbers always results in an Even number When a number (even or odd) is multiplied by an even number, it results in an Even Number. When an even number is subtracted from an odd number or vice-versa, it results in an odd number. ( 7 - 4 = 3 , 8 - 3 = 5 ...) […] 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 […] 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 […] READ MORE...

Tuples in Relations and their examples READ MORE...

A Relation from set A to set B is defined as a set of ordered pairs formed from the elements of set A and B. In other words, a relation is a subset of the cartesian product of sets A and B. The subset is derived by establishing predicate filter(s) or criteria stating a condition that evaluates the qualifying ordered pairs from the cartesian product to be included in the subset as specified by the relation. A relation is uni-directional i.e. if a relation exists from A to B then it does not imply that a relation exists from B to A as well. Also, (a,b) ≠ (b,a) Let a ∈ A and b ∈ B. Let (a,b) be an […] READ MORE...

A function is said to be an even function if the sign of the image does not change when the sign of the preimage changes. Conversely, a function is called an odd function when the sign of the image changes when the sign of the preimage changes. For Even functions, f(x) = f(-x). For the Odd function, f(x) = -f(-x). Examples of Even Function: f(x) = x2 .We have f(1) = 1 and f(-1) = 1 hence f(1) = f(-1).This is true fora ∀ x ∈ R.Another example would be mod function |x|. Examples of Odd Function: f(x) = x3 .We have f(1) = 1 and f(-1) = -1 hence f(1) = -f(-1).This is true for ∀ x ∈ R. READ MORE...