Let S be the set of all real numbers between 0 and 1 with the property that their decimal..

decimal expansions only have 6's and 7's. For example, the following numbers are elements of S: .777766776767777776777... and .6666666666666766667777766666667
show that there exists a rational number in S and show that there exists an irrational number in S. and describe how the number is formed
for the first part can i just say.77777 repeating? and i dont know how to do the second part at all.. thanks!

For a rational number in S, any repeating decimal (with digits 6 and 7) will work.
For instance, 0.777... = 7/9.

For an irrational number, all you need is for the digits not to repeat.
One simple example is 0.676776777677776...

(Remember that the irrationals are those decimals which neither repeat nor terminate.)

I hope this helps!