S={1,{a,c},b,c}?

Hi, can someone help me ou with this, i have no idea how to do it.





Given set S={1,{a,c},b,c}





- P(A) or 2^A





- If the number of proper subsets of a set is 31, then what is the number of elements of this set?





thanks so much:P

S={1,{a,c},b,c}?
If a set S has n elements, there are 2^n (2 to the power of n) different ways to create a subset of S. Look at it this way: Think of S as n light bulbs numbered from 1 to n. If you want a subset X of S, you choose a subset of lights to turn on. So basically, for each light bulb, we choose to turn it on or off, and in the end, the set of light bulbs that we turned on make up the subset X.





For each light bulb we have two states (on or off), and these are all independent. (Choosing to turn one bulb on or off doesn't have any effect on what we do with another bulb.) Therefore, the total number of ways to set the light bulbs is 2x2x2x..x2 (n times), or 2^n.





If we want the number of subsets of S that are not equal to S, the answer is 2^n-1, because one of the subsets we create in the 2^n different subsets is the one where all the light bulbs are on, and therefore the subset is the same as S.





So if a set has 31 proper subsets, the number of elements n is the number n that gives you 2^n-1. Try a few values of n and see which one works!





By the way, the set S={1,{a,c},b,c} has only 4 elements, because {a,c}, being its own separate entity, only counts as one element.





Tell me if you have any more questions.

sweet pea