Set Builder Notations

Question

1) Find 3 different elements that are in both 6Z and 8Z (where Z is an integer). Is there a convenient way to describe the set of all elements that are in both 6Z and 8Z?

 

2) For k is a natural number, define Tk={z is an integer | |z - sq.root of 2| < k}. Find a formula expressing |Tk| in terms of the value of k.

Answer 1

Since Z is Integer, then 6Z=3(2Z) and 8Z=4(2Z)

So the set of all elements in both sets 6Z and 8Z is the set of even. 2Z numbers.

2.Tk={z:|z-√2|<k}

-k<z-√2<k

√2-k<z<k+√2

Tk=(√2-k, √2+k), k=1,2,.. sequence of intervals

(√2-1, √2+1), (√2-2, √2+2), ..

Tk is