Question

For each set of conditions below, give an example of a predicate P(n) deﬁned on N that satisfy those conditions (and justify your example), or explain why such a predicate cannot exist.

(a) P(n) is True for n ≤ 5 and n = 8; False for all other natural numbers.

(b) P(1) is False, and (∀k ≥ 1)(P(k) ⇒ P(k + 1)) is True.

(c) P(1) and P(2) are True, but [(∀k ≥ 3)(P(k) ⇒ P(k + 1))] is False.

(d) P(1) is True, P(k) ⇒ P(k + 1) is False for all k
∈N.

Answer #1

a) is true for only and false for all other naturals

b)

Then is true but P(1) is false

c) so P(1) and P(2) are true but is false

d)

The P(1) is true but is false (one is odd means the other is even)

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For each problem below, either give an example of a function
satisfying the give conditions, or explain why no such function
exists.
(a) An injective function f:{1,2,3,4,5}→{1,2,3,4}
(b) A surjective function f:{1,2,3,4,5}→{1,2,3,4}
(c) A bijection f:N→E, where E is the set of all positive even
integers
(d) A function f:N→E that is surjective but not injective
(e) A function f:N→E that is injective but not surjective

1. For each statement that is true, give a proof and for each
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(a) For all natural numbers n, n2
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(c) For every real number x
³ 1, x2£
x3.
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x^4+ 1/x
-(x+1)^(1/2)=0.
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Open
Sentence: T(x,
y) : xy is even.
Statement:
∃!x∀y T(x, y).
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For all x and y, the product
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< 0}.
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In each case below show that the statement is True or give an
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(i) If {X, Y } is independent in R n, then {X, Y, X + Y } is
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Hint: Find the 0 and 1
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complement operator.

Consider an axiomatic system that consists of elements in a set
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