# (solution) 1. Let U be any set. Prove that for every B (U) there is a

Hi, see doc. attachment and specific notes. Need it today not later than 2 pm, tx.

1. Let U be any set. Prove that for every B ? ?(U) there is a unique D ? ?(U) such that for every C ? ?(U), C B = C ? D. These are my teacher?s notes in which I need to prove the statement above: Thank you. There are some keywords to pay attention to. The first, is ?for every set B in P(U)?. This means we?ll need to let B be in P(U). Similarly, the words ?for every C in P(U)? mean that we need to let C be in P(U) as well. So, to start the problem we?d say, ?Let U be any set, A in P(U), and C in P(U)?. Also, the problem says to show ?there is a unique D in P(U) such that C B = C ? D?. To show this we need to show that this D both exists, AND that it is unique. The most straightforward way to show something exists is to actually construct a specific solution. Since we?re trying to show D exists, let?s propose a specific value for D. Try using D = UB. For this value of D, we need to show that D in P(U), and that C B = C ? D. To show that two sets are equal to each other we typically show that each is a subset of the other. So, in your proof, say ?let x in C B? and then deduce that x in C ? D. Then, say ?let x in C ? D? and deduce that x in C B. These steps establish that for the specific value of D you constructed that C B = C ? D. The final step is to show that the value of D you constructed is unique.