Argue whether or not they will result in choices that satisfy: i) non-emptiness

Question

Argue whether or not they will result in choices that satisfy: i) non-emptiness and ii) choice coherence. If it does then explain why. If it doesnt, then provide a counterexample.

Adrian claims he hates books and says that he has assigned to each bo(k b in X, a negative number h (b) 0 that reflects how much he hates that particular book. However, when taced with the problem of choosng a book from any non-empty set B C X, perversely he selects the book(s) that he hates the most! That is, e)-[oe B:h0)s(0) for all be a)

Explanation / Answer

(1) Non emptiness

A choice function c satisfies finite nonemptiness if c(A) is nonempty for every finite aA

for b B such that a has just one book. b = {b1}

c(B) = {b1 : as h(a) <= h(b) where b B

for b B, such that there are 2 books b1 & b2, b = {b1, b2}

the hate assinged to these books would be h(b1) & h(b2)

there are 3 cases possible

case 1: h(b1) > h(b2) {book 2 is more hated}

c(B) = {b2, as h(b2) < h(b1)

case 2: h(b1) < h(b2) {book 1 is more hated}

c(B) = {b1, as h(b1) < h(b2)

case 3: h(b1) < h(b2) {book 1 and 2 are equally hated}

c(B) = {b1, b2, as h(b1) = h(b2)

So c(B) is never an empty set. Which means choice is resulting into non-emptiness

(2) Choice Coherence

A choice function c satisfies choice coherence if, for every pair x and y from X and A and B from A,

if x, y A B, x c(A), and y c(A), then y c(B)

let

A = {b1, b2, b3}

B = {b2, b3, b4}

A B = {b2, b3}

x, y A B

let x = b2, y = b3

h(b1), h(b2), h(b3) & h(b4) are the hates assigned to b1, b2, b3 and b4

to make x c(A), we assume the following order

(A) h(b2) < h(b3) < h(b1) < h(4)

so c(A) = {b2 as h(b2) < h(b3) < h(b1)

x c(A)

but b3 c(B) because h(b2) < h(b3)

c(B) = {b2}

(B) h(b4) < h(b2) < h(b3) < h(1)

c(A) will be same as h(b2) < h(b3) < h(1)

x c(A)

but c(B) = {b4}

b3 c(B) because h(b2) < h(b3) > h(b4)

hence proved c is choice coherent too.

So choice C satisfies nonemptiness and choice coherence

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