MATHEMATICS ENCYCLOPEDIA

Oxford dictionary definition: An opinion or conclusion formed on the basis of incomplete information. Cambridge dictionary definition: A guess about something based on how it seems and not on proof. Example: Riemann hypothesis, Poincare conjecture, Collatz conjecture etc.

What is the solution of the equation x 2 + 1 = 0? In fact we don’t have any solution to the given equation in the system of real numbers. Why? Because x 2 + 1 = 0 gives x 2 = -1 and square of every real number is non-negative. Then, what could we do? We need to extend the real number system to a larger system to find the solution of the equation x 2 + 1 = 0. In general, we have to find the solution of the equation ax 2 + bx + c = 0, where D = b 2 – 4ac < 0. Euler was the first mathematician to introduce the symbol i ( Iota ) for the square root of -1, i.e. solution of the equation x 2 + 1 = 0. Now, we have i = √-1 i 2 = -1 i 3 = i 2 * i = -i i 4 = ( i 2 ) 2 = ( -1 ) 2 = 1

Definition of linear inequality  A statement involving two real numbers or two algebraic expressions related by the symbol ' < ', ' > ', ' ≤ ', ' ≥ ' form an inequality. Types of inequalities Numerical inequalities, e.g. 7 > 5, 11 < 8 etc. Literal inequalities, e.g. x < 7, y > 4, x ≥  6 etc. Double inequalities, e.g. 3 ≤ x < 5, 6 < y < 11 etc. Strict inequalities, e.g. x < 7, ax + by > 0 etc. Slack inequalities, e.g. ax + by ≤ 0 Quadratic inequalities, e.g. ax^2 + bx + c > 0 etc. Rules for solving an inequality Rule 1: Equal numbers may be added to ( or subtracted from ) both sides of an inequality without affecting the sign of inequality. Rule 2: Both sides of an inequality can be multiplied ( or divided ) by the same positive number. But when both sides are multiplied or divided by a negative number, then the sign of inequality is reversed. Representation of inequality on real number line

Function Function is a special type of relation. It is one of the most important concepts in mathematics.The word function is derived from a Latin word meaning operation and the word mapping and map are synonimous to it. Functions play very important role in Calculus. A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, a function f is a relation from a non-empty set A to a non-empty set B such that the domain of f is A and no two distinct ordered pairs in f   have the same first element . If f is a function from from A to B and ( a, b ) ∈ f , then f  ( a )  = b, where b is called the image of a under f and a is called the preimage of b under f . The function f  from A to B is denoted by f  : A ⟶ B. A function which has either R ( real numbers ) or one of its subsets as its range is called a real valued function. Further, if its domain is also either R ( rea...

Relations A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product         A Х B . The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A Х B. The second element is called the image of the first element. The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the domain of the relation R. The set of all second elements in a relation R from a set A to a set B is called the range of the relation R. The whole set B is called the codomain of the relation R.  Range ⊆ codomain . Example Let A = { 2, 3, 4, 5 } and B = { 3, 6, 7, 10 } . A relation R from set A to the set B as follows: R = { ( x, y ) : x divides y, where x ∈ A and y ∈ B }. We find that 2 divides 6 and 10, 3 divides 3 and 6, 5 divides 10 and there is no number in B that can be divided by 4. Thus ( 2, 6 ) ∈ R, ( 2, 10 ) ∈ R, ( 3, 3...

Cartesian products of sets: If you have two sets A and B. The cartesian product                                                      A Ⅹ B = { (a, b ) : a ∈ A, b ∈ B } Example: Let A = { a, b } and B = { 5, 7, 9 }, then A Ⅹ B = { (a, 5 ), (a, 7 ), ( a, 9 ), ( b, 5 ), ( b, 7 ), ( b, 9 ) }. B Ⅹ A = { ( 5, a ), ( 5, b ), ( 7, a ), ( 7, b ), ( 9, a ), ( 9, b ) }. If either A or B is a null set, then A Ⅹ B will also be empty set, i.e. A Ⅹ B = Φ. Two ordered pairs are equal, if and only if the corresponding first elements are equal and the second elements are also equal. Let two ordered pairs ( a, b ) and ( c, d ) be equal, i.e.,               ( a, b )  = ( c, d ), if and only if a = c and b = d. If there are x elements in A and y element in B, then there are xy elements in A Ⅹ B, i.e., if n(A) = x and n(B) = y...

Axiom Oxford dictionary definition: A statement or preposition which is regarded as being established, accepted, or self-evidently true. Cambridge dictionary definition: A statement or principle that is generally accepted to be true, but need not be so. Example: Peano axiom - For every natural number  n ,  S ( n ) is a natural number. Postulate Oxford dictionary definition: A thing suggested or assumed as true as the basis for reasoning, discussion, or belief. Cambridge dictionary definition : To suggest a theory, idea etc. as a basic principle from which a further idea is formed or developed. Example: Euclid's postulate- A straight line segment can be drawn joining any two points. Theorem Cambridge dictionary definition: A formal statement that can be shown to be true by logic. Oxford dictionary definition: A general preposition not self-evident but proved by a chain of reasoning; truth established by means of accepted truths. Examples: Pytha...