mathematics, machine learning , data science, sets, probability and analytics and logic

1. Algorithmic Foundations
1
.Usingthe laws of logical equivalence show:
(a) ¬ (p∨ (q∨¬r)) ∧q≡ ((¬p ∧ q) ∧ R)
solution…………………………………………………………………………………………………………………
………………………………………………………………
Hence the proof that ¬ (p∨ (q∨¬r)) ∧q is equivalent to ((¬p ∧ q) ∧ R)
(b)(p→r)∨ (q→r) ≡ (p∧q) →r
p q r ((p → r) ∨ (q → r)) p q r ((p ∧ q) → r)
0 0 0 1 0 0 0 1
0 0 1 1 0 0 1 1
0 1 0 1 0 1 0 1
0 1 1
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0 1 1 1
1 0 0 1 1 0 0 1
1 0 1 1 1 0 1 1
1 1 0 0 1 1 0 0
1 1 1 1 1 1 1 1
Hence the proof that (p→r) ∨ (q→r) ∧q is equivalent to (p∧q) →r
(c) (∀×∈⋃ (P(x) →¬Q(x)) ≡ ¬∃×∈⋃. (P(x) ∧Q(x))
P(x) P(x)
0 0
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2. P(x) =
Q(x) =
(P(x) ∧Q(x) =……………………………………………………………………………………………....
Now looking at the truth table P(x) is equivalent Q(x) and (∀×∈⋃ (P(x) →¬Q(x)) is equivalent to ¬∃×∈⋃.
(P(x) ∧Q(x)) hence the proof.
2. Assuming thefollowing predicates:
•L(x, y):x is strictly less than y (x<y);
•E(x):x is even;
•P(x): x is a prime number;
•EQ(x, y):x equals y (x=y);
•G(x):x isgreater than zero (x>O);
•D(x, y): x divides y exactly;
Determine which of the followingformulae are true (in your answer include an expression of
the formula inconcise (good) English without variables).
(a) ∀×∈N. (D(2, x) →E(x))
Given that x/2= is an integer____________________hence x is even hence the proof.
(b) ∃y ∈N. ∀×∈N.L(x, y)
(c) ∃×∈Z+.∀y ∈Z+.(G(x) →(P(y) ∧L(x, y)))
Next,usingtheabove predicates and quantifiers were necessary, express the followingEnglish
statements inlogic.
(d) "Any non-zero integer divides itself"
(e) "Aprime number's only positive factors are 1and itself."
3. Provethat (BA) ⋃ (CA) =(B ⋃C)A using:
(a) A containment proof.
(b) Using set builder notation and logical equivalences.
4. For each of the following functions find the inverse or explain why no inverse exists.
a. f :N→ Nwhere f(x) =4·x2
+ 1
b. g : Z→ Z where g(x) = x + 7
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