mathematical systems

Question 1

≅

Answer 1

congruent geometric figures
(shapes that are exactly the same)

Question 2

it is between 2 graphs means that they have essentially the same structure

Answer 2

isomorphism

Question 3

what is homomorphism

Answer 3

combine 2 edges by deleting a vertex to combine the 2 edge (or soothing out the edges) and repeating the process until u make 2 graphs the same

Question 4

_____ Theorem,
every non-planar graph has a subgraph that is homomorphic to the graphs ___ or ___

Answer 4

K5 or K3,3

Question 5

K5

Answer 5

5 vertices
every vertex is connected to every other vertex

Question 6

K3,3

Answer 6

bipartite graph
2 set of 3 vertices ach. every vertex from one set = form total of 9 edges

Question 7

what are some practical applications of trying to draw a graph as a planar drawing

Answer 7

design of circuit boards (computers & other electronic components) - require wires to arranged in a planar drawing

Question 8

S:
⋆:

Answer 8

S: set
⋆: one binary operation - combine 2 things to form a set to get a new thing (+)

Question 9

what are the 4 properties of a group

Answer 9

1. CLOSURE PROPERTY - sum of two integers is always an integer (For example, 3 + 5 = 8, which is an integer.)
2. ASSOCIATIIVE PROPERTY - the associative property of addition holds true for the integers
   S, a ⋆ b ⋆ c = a ⋆ b ⋆ c
   - associative under addition
   - not associative under subtraction
3. EXISTENCE OF IDENTITY (NEUTRAL)ELLEMENT - no. 0 serves as identity element for addition of integers. cos adding 0 to any integer does not change the integer (For example, 5 + 0 = 5.)
4. EXISTANCEOF INVERSE ELEMENTS - inverse element is -a (For example, the inverse of 3 is -3, since 3 + (-3) = 0.)

we can conclude that the integers form a group under addition

In simple terms,
add integers, we get another integer (closure)
the order of adding numbers doesn’t matter (associativity)
adding 0 doesn’t change the number (identity)
every integer has an opposite that cancels it out (inverse)