MA-131 Key Concepts And Formulae Exam Questions
And Answers 100% Correct
State the 5 axioms of Peano arithmetic (P1-P5). - ANSWER P1=There exists a number 0
(the first number)
P2=Every number has a successor.
P3=0 is NOT the successor of any number ie 0≠s(n) n∈N
P4=If x,y are two numbers such that s(x)=s(y) --> x=y
P5=If P(n) is a statement about number n such that P(0) true, and whenever P(k) true,
then P(k+1) also true. Then P(n) true for all n∈N (PROOF BY INDUCTION)
State addition rules A1 and A2, and also multiplication rules M1 and M2 in Peano
Arithmetic. - ANSWER A1=For any number n, n+0=n
A2=If we have a+b, then a+s(b)=s(a+b)
M1=For all n∈N, n*0=0
M2=If a*b is defined, then a*s(b):a+a*b
Define a Circle Inversion (Hyperbolic Geometry) - ANSWER The circle inversion of Point
A (inside the circle) is a point B such that the points O, A and B are on the same line and
|OA|*|OB|=R²
State the definition of the Poincare disc (Hyperbolic Geometry) - ANSWER The Poincare
Disc D is the disc of radius 1 at the origin.
Define the following terms in Hyperbolic Geometry in relation to the Poincare disc D.
1) P-point
2) P-line
3) P-circle - ANSWER 1) A p-point os a point inside D
2) A p line is a diameter of D OR a circular arc meeting the boundary of D at 90 degrees.
, 3) A p-circle is any circle inside D.
State Lemma 3.3 in Hyperbolic Geometry - ANSWER Given 3 non-collinear points A,B,C
(meaning that they are not on a straight line), there exists a unique circle through them.
State the three distinct universes of Geometry in relation to parallel lines - ANSWER
HYPERBOLIC: More than 1 parallel exists
EUCLIDEAN: Unique parallel exists
ELLIPTIC: No parallels at all.
State the 5 axioms of ELLIPTIC GEOMETRY - ANSWER 1) Any 2 points can be joined by a
straight line (the shortest distance).
2) Any straight line segment can be extended indefinitely in a straight line.
3) Given any straight line segment on a sphere, a circle can be drawn having the
segment as radius and one endpoint as centre (the line of latitude makes a circle).
4) All right angles are congruent.
5) There are NO parallel lines.
Define a SACCHERI quadrilateral for EUCLIDEAN, ELLIPTIC and HYPERBOLIC
geometry. - ANSWER EUCLIDEAN: A quadrilateral with base AB,
angleA=angleB=90,|BC|=|AD|.
ELLIPTIC: A quadrilateral with base AB, angleA=angleB=
obtuse,|BC|=|AD|.
HYPERBOLIC: A quadrilateral with base AB, angleA=angleB=acute,|BC|=|AD|.
Define the UPPER HEMISPHERE - ANSWER The upper hemisphere is a set given by
{(x,y,z)∈R³|x²+y²+z²=1,z>0}
State Facts 1-4 in Elliptic Geometry - ANSWER 1) Given 3 points in R³ either they are all
on the same line or they determine a unique plane.
And Answers 100% Correct
State the 5 axioms of Peano arithmetic (P1-P5). - ANSWER P1=There exists a number 0
(the first number)
P2=Every number has a successor.
P3=0 is NOT the successor of any number ie 0≠s(n) n∈N
P4=If x,y are two numbers such that s(x)=s(y) --> x=y
P5=If P(n) is a statement about number n such that P(0) true, and whenever P(k) true,
then P(k+1) also true. Then P(n) true for all n∈N (PROOF BY INDUCTION)
State addition rules A1 and A2, and also multiplication rules M1 and M2 in Peano
Arithmetic. - ANSWER A1=For any number n, n+0=n
A2=If we have a+b, then a+s(b)=s(a+b)
M1=For all n∈N, n*0=0
M2=If a*b is defined, then a*s(b):a+a*b
Define a Circle Inversion (Hyperbolic Geometry) - ANSWER The circle inversion of Point
A (inside the circle) is a point B such that the points O, A and B are on the same line and
|OA|*|OB|=R²
State the definition of the Poincare disc (Hyperbolic Geometry) - ANSWER The Poincare
Disc D is the disc of radius 1 at the origin.
Define the following terms in Hyperbolic Geometry in relation to the Poincare disc D.
1) P-point
2) P-line
3) P-circle - ANSWER 1) A p-point os a point inside D
2) A p line is a diameter of D OR a circular arc meeting the boundary of D at 90 degrees.
, 3) A p-circle is any circle inside D.
State Lemma 3.3 in Hyperbolic Geometry - ANSWER Given 3 non-collinear points A,B,C
(meaning that they are not on a straight line), there exists a unique circle through them.
State the three distinct universes of Geometry in relation to parallel lines - ANSWER
HYPERBOLIC: More than 1 parallel exists
EUCLIDEAN: Unique parallel exists
ELLIPTIC: No parallels at all.
State the 5 axioms of ELLIPTIC GEOMETRY - ANSWER 1) Any 2 points can be joined by a
straight line (the shortest distance).
2) Any straight line segment can be extended indefinitely in a straight line.
3) Given any straight line segment on a sphere, a circle can be drawn having the
segment as radius and one endpoint as centre (the line of latitude makes a circle).
4) All right angles are congruent.
5) There are NO parallel lines.
Define a SACCHERI quadrilateral for EUCLIDEAN, ELLIPTIC and HYPERBOLIC
geometry. - ANSWER EUCLIDEAN: A quadrilateral with base AB,
angleA=angleB=90,|BC|=|AD|.
ELLIPTIC: A quadrilateral with base AB, angleA=angleB=
obtuse,|BC|=|AD|.
HYPERBOLIC: A quadrilateral with base AB, angleA=angleB=acute,|BC|=|AD|.
Define the UPPER HEMISPHERE - ANSWER The upper hemisphere is a set given by
{(x,y,z)∈R³|x²+y²+z²=1,z>0}
State Facts 1-4 in Elliptic Geometry - ANSWER 1) Given 3 points in R³ either they are all
on the same line or they determine a unique plane.
