Unit-I: Sets and Functions
Unit-II: Algebra
Unit-III: Coordinate Geometry
Unit-IV: Calculus
Unit-V: Mathematical Reasoning
Unit-VI: Statistics and Probability

Space and its dimension

A space is said to be of dimension zero, one, two or three as it comprises of a single  point, line, plane or contains points not all of which are co-planers.

 \to ‘Space’ shall mean a three -dimensional space.

Some Axioms on Space

(i) Axiom-1:

Space is a non-empty set of points.

(ii) Axiom -2:

If  P \text {and } Q are distinct points in space  S then \overleftrightarrow{PQ} \text { CS} and if   \lambda is a plane containing  P \text { and } Q then  \overleftrightarrow{PQ} \text { C } \lambda \text { CS }

(iii) Axiom -3:

Given any three non-collinear point  P \text { and } Q  in space  S is exactly one plane  \lambda  such that \{ P, Q, R \}  \text { C } \lambda  \subset \text { S}

(iv) Axiom -4:

      If \lambda_1 \text { and } \lambda_2 are two distinct planes in a space and  P  is a point such that  P \in \lambda_1 \cap \lambda_2 then there exists another point   Q  different from  P such that  Q \in \lambda_1 \cap \lambda_2

CONVEX SET

A subset  A of a space  S is said to be a Convex set if, for all  P, Q \in A, \overline{PQ} \subset A . Otherwise it is not a convex set.

 A is a Convex Set
B  is not a Convex set, since not all the point on  \overline{P'Q'} is in  B i.e.,  \overline{P'Q'} \not\subset \text {B}

(v) Axiom -5:

     If  \lambda is a plane is space  S then the point of  S not contianed in  \lambda are divided into two disjoint non-empty convex sets  \text {S}_1 \text { and } \text {S}_2  such that   P \in \text {S}_1, \quad Q \in \text {S}_2 \implies \overline{PQ}\cap \lambda \neq \phi

 \text{S}_1 \text { and } \text {S}_2 are called two sides of the plane.

Note:

If a point belongs to space we also say that the point lies on/in that space.

 P is on the space  S  .  Q in the space  S.

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