Question

is anybody good at discrete math?

Answer 1

me

Answer 2

Thanks!!

Answer 3

alright my question is how do you figure out if anything is antisymmetric

Answer 4

thats not very specific

Answer 5

like okay R1 (1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4)

Answer 6

In mathematics, a binary relation R on a set X is antisymmetric if, for all a and b in X

if aRb and bRa then a = b,

Answer 7

yeah but how does that explain if this is antisymmetric or not

Answer 8

well first you have to state what the relation is, equality? less than, greater than? division modulo n?

Answer 9

I would say this is not anti-symmetric because you have
(3,4) ( 4,1) but 3 does not equal 1

Answer 10

if you delete the point (3,4) then it is anti-symmetric

Answer 11

what about (1,2) and (2,1) and (1,1)

Answer 12

Wait so basically its anti symmetric if no points equal each other like a is not equal 2 b?

Answer 13

1R2 and 2R1 so 1 =1 is true

Answer 14

ohh so in that situation a = b?

Answer 15

and if thats the case then it would be antisymmetric

Answer 16

well the order relation <= is anti-symmetric

Answer 17

so the definition of anti-symmetry generalises what we find in the usual order relation <= .
for instance if a<=b and b <= a , then a = b , where a,b are integers and <= means less than or equal (usual ordering).
but notice you can have unusual relations, like subset.
if A is a subset of B and B is a subset of A, then A = B

Answer 18

i dnt get that sorry this type of math is really confusing it took me 3 weeks to understand sets and now im trying to understand relations bim pretty much behind in class :/

Answer 19

That make sense but how does that apply to that problem

Answer 20

well we try to find in math the most general or abstract definitions. we find definitions that strip the terms of any specific meaning, so we can generalise to many different types of relations.

Answer 21

so we find many 'things' that behave in math just like the ordering <= where 2<= 3 . in advanced math we look for patterns

Answer 22

Yeah thats why im being confused because im so used to specific examples that this abstract way of thinking leaves me lost in the dark

Answer 23

ohhh okay

Answer 24

so how do you define <= . when you define it, you realise that other mathematical objects behave similiar to how 2 <= 3 . for instance {1,2} is a subset of {1,2,3}

Answer 25

Yeah
i get that

Answer 26

so you could say {1,2} < {1,2,3} , or you could say <= to indicate that it is not a proper subset

Answer 27

alright

Answer 28

so using our austere definition we have R is antisymmetric
if whenever (a,b) is in R and (b,a) is in R, then it must be the case that a = b

Answer 29

OHHHH IGET IT !!!!!!!

Answer 30

lol thanks but hold on
lol hold on

Answer 31

lol

Answer 32

right, aRb is the same thing as saying (a,b) is in R. or equivalent

Answer 33

wait but if thats the case where (a,b) element of R and (b,a) an element of R and a = b then isnt that kinda like being symmetric

Answer 34

because (1,2) is symmetric to (2,1)

Answer 35

symmetric is different

Answer 36

how so

Answer 37

R is symmetric when if (a,b) is in R, then (b,a) is in R

Answer 38

thats the same thing i said tho

Answer 39

antisymmetry is different

Answer 40

right, sorry i didnt see what you wrote

Answer 41

yeah but using that.. isnt that the same as antisymmetric

Answer 42

ok so here are some examples one sec , thinking

Answer 43

okay

Answer 44

R = { (1,1) (2,1) (2,1) } is antisymmetric.
R= { (1,1) , (1,2) (2,1) } is symmetric

Answer 45

A relation can be both symmetric and antisymmetric

Answer 46

woops

Answer 47

R= { (1,1) ( 1,2) } is anti symetric

Answer 48

a null set would be everything but i dnt get why the first one is antisymmetric

Answer 49

a = 2 right?

Answer 50

and b = 1?

Answer 51

lol careful you are speaking to an open mind at the moment

Answer 52

R= { (1,1) ( 2,1) } is antisyemmetric

Answer 53

The terms 'symmetric' and 'anti-symmetric' apply to a binary relation R:

R symmetric means: if aRb then bRa.
R anti-symmetric means: if aRb and bRa, then a=b.

Answer 54

okay i get that now because a in the R = {(1,1), (2,1)} is basically 1 and b is 1 as well
so a is 1 in the (1,1) and b is 1 in the (2,1) right?

Answer 55

well it satisfies it because it doesnt violate it

Answer 56

it satisfies it trivially, since there is no occurence of both (a,b) and (b,a) in R
R = {(1,1), (2,1)}

Answer 57

the antisymmetry is an 'AND' condition. if you have both (a,b) AND (b,a) , then a=b.
the symmetry condition is if-then condition

Answer 58

hmm i guess sheesh why is this math so hard i swear who thinks of creating stuff like this

Answer 59

well they are both if then conditions, but the antisymmetry has a little extra in it

Answer 60

okay so symmetry mean something must happen first then something else happen.. its like a program

Answer 61

R is symmetric means: if (a,b) is in R then (b,a) is in R.
R anti-symmetric means: if (a,b) and (b,a) is in R, then a=b.

Answer 62

okay so what about this one (3,4) what properties you give this
in my eyes i see no reflexive no symmetry no antisymmetry and no transitive

Answer 63

yes its like rules for a checking program , you could program it to check

Answer 64

okay

Answer 65

so just that one point?
R = { ( 3,4) } ?

Answer 66

yea

Answer 67

then it is not symmetric, but it is anti-symmetric (trivially)

Answer 68

its in an example in my book i hate that they dnt explain stuff

Answer 69

Yes thats what they said but it makes no sense

Answer 70

it is anti symetric why? do you see, because it is an if then condition

Answer 71

how its only (3,4)

Answer 72

if (a,b) and (b,a) are in R, then a=b. but you dont have (a,b) and (b,a) in R

Answer 73

yea okay you just have (a,b)

Answer 74

because 'if-then' is true when the 'if' part is false

Answer 75

i think thats called trivial or vacuously true. we just say true though :)

Answer 76

are you talking about the logic statement if p is false but q is true then the whole statement ends up being true?

Answer 77

p-> q is true when p is false.

Answer 78

or another way to read the definition... is
,

Answer 79

Program:
if you can find two points in your relation (a,b) & (b,a) .... then it must be the case that
a = b.
else true

Answer 80

or true otherwise

Answer 81

and a specific example would be (3,4) so basically any single point except (1,1) or other any points similar to that are antisymmetric

Answer 82

Program: if you can find two points in your relation (a,b) & (b,a) .... then it must be the case that a = b. (if a does not equal B then your anti-symmetry is false.)
otherwise true (if you cant find any two points (a,b) (b,a) )

Answer 83

right , any relation
R = { (a,b) } is antisymmetric , trivially , since there is only one point

Answer 84

since you need at least two points to even check the condition

Answer 85

true cool so one answer down for the rest of my life a single point is antisymmetric

Answer 86

lol

Answer 87

well, you have to be more precise.
the relation R= { (a,b)} is anti symmetric

Answer 88

and also better say its a binary operation, and 'point' is an element of your relation aRb or (a,b) ordered pair

Answer 89

yeah lol that .. R = { (a,b) }

Answer 90

now, how can we make your original relation antisymmetric. can we delete something

Answer 91

okay i get you

Answer 92

R= { (1,1),(1,2),(2,1),(2,2),(3,4),(4,1),(4,4) }
now what are the least amount of points we can delete to make this anti-symmetric
its no fair if we delete all but one , lol

Answer 93

lol okay

Answer 94

all try my best

Answer 95

ill*

Answer 96

ask yourself, why does it fail anti-symmetry (clue)

Answer 97

(4,40 and (1,1) and (2,2) and (3,4)

Answer 98

(4,4) i meant