### Reflexive

iff Ax((x,x) eR)

e.g. S={1,2,3,4}, R = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (3,3), (3,4), (4,4)} (i.e., the <= relation, is reflexive)

### Irreflexive

iff Ax((x,x) !eR)

e.g. S={1,2,3,4}, R = {(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)} (i.e., the < relation, is irreflexive)

### Symmetric

iff AxAy((x,y)eR -> (y,x)eR)

e.g. S={1,2,3,4},: R={(2,1), (1,2), (2,3), (3,4), (3,2), (4,3)}

This relation is "elements whose absolute difference is 1."

### Antisymmetric

iff AxAy((x,y)eR /\ (y,x)eR) -> (x=y)

e.g. S={1,2,3,4}, R = {(1,1), (1,2), (1,3), (1,4), (2,2), (2,3), (2,4), (3,3), (3,4), (4,4)} (i.e., the <= relation)

### Asymmetric

iff AxAy((x,y)eR -> (y,x)!eR)

E.g., the relation < ("less than") = {(1,2), (1,3), (1,4), (2,3), (2,4), (3,4)}

### Transitive

iff AxAyAz(((x,y)eR /\ (y,z)eR) -> (x,z)eR)

E.g., <=, <, >=, > (think Hypothetical Syllogism)

### Implications of Properties

Reflexive -> !Asymmetric

Asymmetric -> Antisymmetric

!Antisymmetric -> !Asymmetric

### Solving Linear Homogeneous Recurrence Relations

r^2 - c(sub1) r - c(sub2) = 0

a(sub1)r^n + a(sub2)r(sub2)^n

### Solving Linear Homogeneous Recurrence Relations with a single root

r^2 - c(sub1)r - c(sub2) = 0 produces two rs of same val.

a(sub1)r^n + a(sub2)nr^n