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.

$B$ 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.

$\text{S}_1 \text { and } \text {S}_2$ 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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