It is not clear if you have learned the inclusion/exclusion principle. If you have, you can apply it without drawing the Venn diagram.
However, for a two-set problem like this, it is easier to draw a Venn diagram and solve accordingly.
There are three sets in a universal set E where the cardinality (i.e. total number of elements) |E|=35.
We are also given that for the set of green hats G, |G|=20.
We conclude therefore that for the set R, |R|=|E|-|G|=35-20=15.
Of the 35 hats, irrespective of colour, 18 of them have logos, so belong to the set L, where |L|=18.
We are required to find the set of hats which are either red, or has a logo, that is, the cardinality of the set R∪L, or |R∪L|.
Consider now the sets L and R.
Draw a Venn diagram for the two, with an intersection, i.e. both red and have a logo. We understand that |L∩R|=11.
So if you put in the Venn diagram 18 for L, 15 for R, and 11 for R∩L. You can calculate that L-R (i.e. with a logo but not red) is 18-11=7, and R-L (i.e. red but no logo) is 15-11=4.
|R∪L| = 7+11+4 = 22.
Thanks a lot.