Question

In each part below, give a formal proof that the sentence given is valid or else provided an interpretation in which the sentence is false.

(a) [∀xA(x) ∨ ∀xB(x)] → ∀x[A(x) ∨ B(x)].

(b) [∃xA(x) ∧ ∃xB(x)] → ∃x[A(x) ∧ B(x)].

Answer #1

In each part below, give a formal proof that the sentence given
is valid or else provided an interpretation in which the sentence
is false.
(a) ∀xA(x) → ∃x[B(x) → A(x)].
(b) ∃x[B(x) → A(x)] → ∃xA(x).

Please explained using formal proofs in predicate logic
In each part below, give a formal proof that the sentence given
is valid or else provided an interpretation in which the sentence
is false.
(a) ∀xP' (x) → ∃x[P(x) → Q' (x)].
(b) ∃x[P(x) → Q' (x)] → QxP' (x).

Please explained using the inference rules and also show all
steps.
In each part below, give a formal proof that the sentence given
is valid or else provided an interpretation in which the sentence
is false.
(a) ∀xP' (x) → ∃x[P(x) → Q'(x)].
(b) ∃x[P(x) → Q'(x)] → ∃xP' (x).

Please explained using the inference rules and also show all
steps. In each part below, give a formal proof that the sentence
given is valid or else provided an interpretation in which the
sentence is false.
(a) ∀xP' (x) → ∃x[P(x) → Q'(x)].
(b) ∃x[P(x) → Q'(x)] → ∃xP' (x).

3. Transform each informal argument into a formalized wff. Then
give a formal proof of the wff. (a) Every student likes cake and
likes ice cream. Fred is a student. Therefore, some student likes
cake and likes ice cream. (b) Every even number is divisible by 2.
There is an even number. Therefore, there is a number which is
divisible by 2.

For each part below, give an example of a linear system of
three equations in three variables that has the
given property. in each case, explain how you got your answer,
possibly using sketches.
(a) has no solutions
(b) has exactly one solution which is (1, 2, 3).
(c) any point of the line given parametrically be (x, y, z) = (s
− 2, 1 + 2s, s) is a solution and nothing else is.
(d) any point of the...

1. For each statement that is true, give a proof and for each
false statement, give a counterexample
(a) For all natural numbers n, n2
+n + 17 is prime.
(b) p Þ q and ~ p Þ ~ q are NOT logically
equivalent.
(c) For every real number x
³ 1, x2£
x3.
(d) No rational number x satisfies
x^4+ 1/x
-(x+1)^(1/2)=0.
(e) There do not exist irrational numbers
x and y such that...

For each of the following statements: if the statement is true,
then give a proof; if the
statement is false, then write out the negation and prove that.
For all sets A;B and C, if B n A = C n A, then B = C.

For each part below, use a complete sentence to comment on how
the value obtained from the data compares to the theoretical value
you expected from the distribution X ~ U(0, 1).
For reference, the empirical data is as follows: Mean = .5503
Standard Deviation = .2729, 1st Quartile = .3525 Median = .6106,
3rd Quartile = .7276
minimum value:
third quartile:
1st quartile:
maximum value:
median:
width of IQR:
overall shape:

For each of the statements below, say what method of proof you
should use to prove them. Then say how the proof starts and how it
ends. Pretend bonus points for filling in the middle.
a. There are no integers x and y such that x is a prime greater
than 5 and x = 6y + 3.
b. For all integers n , if n is a multiple of 3, then n can be
written as the sum of...

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