(1) What is: (a) N ∪ {n ∈ N|n is even}? (b) {a, b, {1, 2}} ∩ {a,...

Question
(1) What is:
(a) N ∪ {n ∈ N|n is even}?
(b) {a, b, {1, 2}} ∩ {a, 1, {2}}?
(c) {n ∈ N|0 > n}?
(2) Say whether each of the following functions is an injection (one-one), surjective (onto), bijective (both one-one and onto), or neither injective nor surjective. You do not need to show any work:
(a) Let f : X → Y, where X = {a, b, c, d, e}, Y = {d, e, g, h, i}:
f = {(a, d), (b, e), (c, g), (d, h), (e, i)}
(b) Let f : N × N → N, f (n,m) = n + 2m
(c) Let f : N → N, f (n) = 3n
(3) Show that equinumerosity is an equivalence relation. (Recall, sets A and B are equinumerous if there exists a bijection between A and B. You need to check that this relation is reflexive, transitive, and symmetric.)
(4) Show that for any sets A, B, and C, (A ∩ B) ∪ C = A ∩ (B ∪ C) iff C ⊆ A.
(5) Let f : X → Y for some arbitrary sets X and Y, and let A ⊆ X and B ⊆ Y.
Show that:
(a) If f is injective, f −¹ [f [A]] = A
(b) If f is surjective, f [f −¹ [B]] = B
(c) If f is not injective, there exists A ⊆ X such that (a) fails.
(d) If f is not surjective, there exists B ⊆ Y such that (b) fails.
(6) Let A ∼ B mean that A and B are equinumerous (i.e., there exists a bijection between them). Show that:
(a) A × B ∼ B × A
(b) If A − B ∼ B − A then A ∼ B (where X − Y = {x ∈ X|X ∉ Y})
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1. a. ℕ ∪ {n ∈ ℕ |n is even}?
Answer: ℕ
b. {a, b, {1, 2}}⋂ {a, 1, {2}}
Answer: a
c. {n∈ ℕ|n
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        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
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        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
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        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
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        Backtrace
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        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
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