SETS\ Let

T

be a set, let

x

be a set of subsets of

T

, and let

Y={S^(C)subeT:Sinx}

. Show that the mapping

f

that maps each

Sin

x

to

S^(C)inY

is a bijection.\ Prove the following claim by way of contradiction. For every set

S

, if

|S|

, then for every two subsets

T,RsubeS

such that

T

and

R

are disjoint it holds that

|T|+|R|

.\ When proving the claim, state explicitly and fully what is the assumption that you are making towards a contradiction.\ (This question is intended to hammer in the point about proofs by contradiction that was explained in Piazza.)\ Recall that, as we learned in class, if

S_(1)

and

S_(2)

are disjoint sets, then

|S_(1)\\\\cup S_(2)|=|S_(1)|+|S_(2)|

. Prove a generalization for multiple sets: If

n>=2

sets

S_(1),S_(2),dots,S_(n)

are pairwise-disjoint (i.e., for each distinct

i,jin[n]

it holds that

S_(i)

and

S_(j)

are disjoint), then\

|S_(1)\\\\cup S_(2)\\\\cup dots\\\\cup S_(n)|=\\\\sum_(iin[n]) |S_(i)|

\ (Inclusion-exclusion principle, special case.) Prove that for any two sets

S

and

T

it holds that

|S\\\\cup T|=|S|+|T|-|S\\\\cap T|

1. Let T be a set, let X be a set of subsets of T, and let Y={SCT:SX}. Show that the mapping f that maps each S X to SCY is a bijection. 2. Prove the following claim by way of contradiction. For every set S, if S