September example

Question 1

Εquivalence relations example

Answer 1

1) X set of all students in MATH 340 R relation such that
aRb <=> a,b were born in the same month

2) X any set R={(a,a)|a∈X}⊆X*X (aRb <=>a=b)

3)X any set R={(a,b)|a,b∈X}=X*X

4) X=Z m∈|N

aRb <==>b-a is divisble by m
All equi relations

Question 2

If
1. x.y=x.z then y=z
2. y.x=z.x then y=z

Answer 2

Proof #1

Since g is a Group of some W belongs to G such that w.x=e

(w.x).y=w.(x.y)=w.(x.z)=(w.x).z

we have w.x=e

therefore y=z

Question 3

Example of rings

Answer 3

(Z,+,) is a ring
(Q,+,) is a ring
(R,+,) is a ring
(MatR(n,n),+,) is a ring

Question 4

Example of ring without unity

Answer 4

2Z={n∈Z|2/n}
(+,*) give BO on 2Z

+: 2a+2b=2(a+b)
:2a2b=2(a*b)

Question 5

Examples of Subring

Answer 5

n∈Z, nZ={a∈Z|n/a} is a subring of Z

Z⊆Q⊆R, z is subring of Q, Q is subring of r

Question 6

Examples of BINARY RELATIONS

Answer 6

X=Z, {(a,a)|a∈Z}
aRb <==>a=b

{(a,b)|a>b}
aRb <==>a>b