Plan II Math Exam 2

one-to-one

What does it mean for two sets to have the same cardinality? There exists a ___-__-___ correspondence between their elements.

2n, even numbers

Show that there is one-to-one correspondence between N and the even numbers.

f(n) = ___ is a bijection from natural numbers to ____ ____

listing, one-to-one

How do we know the integers Z have the same cardinality as N?

By _____ them in ___-___ order

Countable, diagonal

Are rational numbers countable or uncountable?

_____- they can be listed using _____ traversal of a grid.

five

How many platonic solids are there?

Tetrahedron, cube, octahedron, dodecahedron, icosahedron

Name the five platonic solids:

The octahedron

What is the dual of the cube?

congruent, vertex

What makes a solid a platonic solid?

All solids are _____ regular polygons, and the same number meet at each ____.

16

How many vertices does a 4D cube have?

32

How many edges does a 4D cube have?

perpendicular00

How can we understand the 4th dimension?

As a new direction _____ to all 3D space directions, via projections or analogies.

V - E + F = 2

What is Euler’s formula for polyhedra?

only five

What does Euler’s formula help prove?

There are ____ ____ platonic solids

Convex polyhedra

On what kind of shapes does Euler’s formula apply?