Two Discrete Math Question?

Question

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Pls help

Answer 1

In the first answer, I'm assuming Φ is meant to be the empty set. If not, please indicate.


1.

(i) P(C), the power set of C, is the set of all subsets of C. The subsets of C are:

Φ,
{{Φ}},
{{{Φ}}},
and {{Φ}, {{Φ}}} (= C). Thus

P(C) = {Φ, {{Φ}}, {{{Φ}}}, {{Φ}, {{Φ}}}}.

ii) ∪ S where the union is taken over all elements S &isin: C is the set which comprises all elements x for which ∃ S ∈ C with x ∈ S. More succintly

∪ S = {Φ} ∪ {{Φ}} = {Φ, {Φ}}.

2.

First we show that ⊕ is well defined. For [x], [y] ∈ J_3, with x and x' ∈ [x], and y and y' ∈ [y], we need to show that

[x + y] = [x' + y'],

ie. that our choice of equivalence class representative does not change the value of [x]⊕[y]. Now,

x - x' = 3j,
y - y' = 3k,

for some integers j and k. Then

(x + y) - (x' + y') = (x - x') + (y - y') = 3j + 3k = 3(j + k).

Thus (x + y) - (x' + y') is divisible by 3 and so [x + y] = [x' + y']. Hence ⊕ is well defined.

Now we show ⊕ is onto. Simply observe that for any [x] ∈ J_3, that

[x] ⊕ [0] = [x + 0] = [x],

and so the image of ⊕ is the whole of J_3, ie. ⊕ is onto.

Finally we need to show that is not one-to-one. To do this, we need to find elements [x], [y], [j], [k] ∈ J_3 with either [x] ≠ [j] or [y] ≠ [k] and

[x] ⊕ [y] = [j] ⊕ [k].

This is simple, for [1] ≠ [0], [2] ≠ [0], and yet

[1] ⊕ [2] = [1 + 2] = [3] = [0]

(because 3 - 0 = 3)

= [0 + 0] = [0] ⊕ [0].

This concludes our business here.

Answer 2

R U S = { a million,a million ; a million, 2 ; 2,a million ; 2, 2 ; 2,3 ; 3,a million ; 3, 2 ; 3,3 ; 3, 4 } = S a million) means set distinction. a extra robust thank you to call it's going to be "not including", considering that set distinction could be slightly confusing for you. as an occasion A B means each and all of the climate in A that are actually not in B. 2) i'm undecided what this suggests. 3) that's often the set cartesian product. although, it could have many distinctive meanings, gazing the definition given in the particular question. 4) n is the set intersection. as an occasion, R n S means each and all of the climate that are in the two R and S.