Mathematics Test - 12

Consider a point P in space. The position of the point P is given by triad (x, y, z) where x, y, z are sensed(i.e. algebraic) distance from YZ-plane, ZX-plane and XY-plane respectively.

If an event is impossible it’s probability is zero. If an event is certain to occur it’s probability is 1.The probability of any other event is between these two values.

Main Concept :
Tautology in mathematical reasoning

Tautology: A statement pattern is called a tautology, if it is always true, whatever may be the truth values of constitute statements.

A tautology is called a theorem or a logically valid statement pattern. A tautology is always true for all possible truth values of its component statements.

Other Concepts :

Concept 1 :
Negation of a statement in mathematical reasoning

The denial of a statement p is called its negation, written as ~ p. Negation of any statement p is formed by writing "It is not the case that ..... "or " It is false that......." before p or, if possible by inserting in p the word "not".

Negation is called a connective although it does not combine two or more statements. In fact, it only modifies a statement.

Negation of compound statements:
We have learnt about negation of a simple statement. Writing the negation of compound statements having conjunction, disjunctions, implication, equivalence, etc, is not very simple. So, let us discuss the negation of compound statement.

Concept 2 :

Simple & Compound statements in mathematical reasoning

(i) Simple statements: Any statement or proposition whose truth value does not explicity depend on another statement is said to be a simple statement.

In other words, a statement is said to be simple if it cannot be broken down into simpler statements, that is, if it is not composed of simpler statements.

(ii) Compound statements: If a statement is combination of two or more simple statements, then it is said to be a compound statement or a compound proposition

Plane A plane is a surface such that, if any two points are taken on it, the line segment joining them lies completely on the surface.

Equation of Plane in Different Form
(i) General Equation of a Plane

General equation of a plane is ax + by + cz + d = 0

(ii) Equation of a Plane Passing through a Given Point

The general equation of a plane passing through a given point (x₁, y₁, z₁) is a (x - x₁) + b(y - y₁) + c(z - z₁) = 0, where a, b, c are direction ratios of a line perpendicular to plane. 

(iii) Intercept Form of the Equation of a Plane

(iv) Equation of Plane in Normal Form

The equation of a plane in normal form is lx + my + nz = p where l, m and n are direction cosines of a line normal to plane and p is perpendicular distance of the plane from origin.