Discrete Math Worked Problems

7/25/2019 Discrete Math Worked Problems

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theorem 3.2.1

every equiv class is a subset of a...

proove b n pg 150



THEOREM 3.2.1a) x/R means:

the set of all y's belonging to set A such that x is related to y.

the set of all y's belonging to A such that x is related to y is a subset of A and xbelongs to the set of all y's belonging to A such that x is related to y. Thus everyequivalence class is a nonempty subset of A.(it's got it's x's and it's got it's y's)

b) x is related to y if and only if every y belongs to set A such that x is related toy EQUALS every x belonging to A such that y is related to x. Thus the elementsare related if and only if their equivalence classes are identical.

(This could mean all the x's and y's are the same, or it could mean that whicheverway you go, the relationship is defined the same: eg John is george's brother,George is John's brother)

c) x is not related to y if and only if all the y's in set A such that x is related to yor all the x's in A such that y is related to x results in an empty set. Thuselements in A are unrelated if and only if their equivalence classes are disjoint.(x and y aren't related if one of them is an empty set)

3.5

GRAPHS....

Graphs:A graph G is a pair(V,E) where V is a nonempty set and E is a set of unorderedpairs of distinct elements of V.

A Vertex is a point.an Edge is the line between the points.

if you see 'uv' that means the edge (line) between u and v.



Isomorphic: Two different looking graphs, but all the points have the same edgesgoing to them.....basically the same graph, but looking different (eg, me andsomebody else both draw a graph of something and we do it right, but they lookdifferent. Maybe I do a above b, and he does b to the right of a or whatever....)

*Can also name the points differently!!!

the ORDER of G means "How many points?"the SIZE of G means "How many lines?"ADJACENT.......

YO, TO PROVE SOMETHING, TRY WRITING IT IN PLAIN English, then turningit to mathese...... just an idea...

u belongs to uv if there's an edge between them.

MATHESE:

two points are ADJACENT if there's a line between them.MATHESE:

"Two vertices u and v a adjacent iff uv belongs to the edge!"

The DEGREE of a point (vertex) is how many lines go to it.

a NULL GRAPH has only points and no lines.

COMPLETE GRAPH: Where every pair of distinct points has a line betweenthem. It'll have one-less amount of lines going to the points than the amount ofpoints in the graph.

HANDSHAKING LEMMA:a) The number of lines going to each point is twice the amount of lines in thegraph.b) The number of points that have an odd number of lines going to it is even.



SUBGRAPHS:

Parts of a graph....

WALK: Just like it says, start at any point and cruise to the next and to the nextand so on....

The LENGTH of the walk is how many edges are counted along the journey(same way we'd count it even if we didn't know any of this)

A PATH is a walk where all points (vertices) are unique, except the beginning andthe end (same as any old path you can think of). It can be circular.

a walk is CLOSED if we ended where we started.

A graph G is CONNECTED if every vertex is reachable from anothervertex ....Connected (the dots are connected)..... disconnected otherwise

NOTE: A single point (with no lines) counts as well!

Discrete Math Worked Problems

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