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I am Lyosha [343]
3 years ago
7

The symbol ⊕ is used to denote exclusive or, so p ⊕ q ≡ (p v q) ^ ~ (p ^ q).

Mathematics
1 answer:
Elan Coil [88]3 years ago
3 0

I'll use text forms of the logical symbols because copying/pasting on mobile is annoying.

⊕ = XOR

∨ = OR

∧ = AND

~ = NOT

↔ = IFF (meaning "if and only if")

→ = IMP (short for "implies")

and for logical equivalence I'll use a regular equal sign.

Then

p XOR q = (p OR q) AND NOT (p AND q)

(a) By definition of XOR, we have

p XOR p = (p OR p) AND NOT (p AND p)

= p AND NOT p

and this is tautologically FALSE.

(b) To rewrite XOR, first distribute the NOT over the AND using DeMorgan's law:

NOT (p AND q) = NOT p OR NOT q

Then we can interchange the AND and ORs using their respective distributive properties. That is,

p XOR q = (p OR q) AND (NOT p OR NOT q)

= (p AND (NOT p OR NOT q)) OR (q AND (NOT p OR NOT q))

= ((p AND NOT p) OR (p AND NOT q)) OR ((q AND NOT p) OR (q AND NOT q))

Both (p AND NOT p) and (q AND NOT q) are tautologically FALSE, so the truth value of either OR depends only on the other statements, so we have

p XOR q = (p AND NOT q) OR (q AND NOT p)

Recall the definition of IFF:

p IFF q = (p IMP q) AND (q IMP p)

Also recall that IMP is logically equivalent to

p IMP q = p OR NOT q

Then we have

p IFF q = (p OR NOT q) AND (q OR NOT p)

so that its negation is

NOT (p IFF q) = NOT ((p OR NOT q) AND (q OR NOT p))

= NOT (p OR NOT q) OR NOT (q OR NOT p)

= (NOT p AND q) OR (NOT q AND p)

The statements in bold match, so p XOR q is indeed equivalent to NOT (p IFF q).

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pav-90 [236]

if the 20% of a number is 386, what will be 30% of the same number

Answer will be 157

4 0
3 years ago
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Find the absolute maximum and absolute minimum values of f on the given interval. f(x) = 6x3 ? 9x2 ? 108x + 2, [?3, 4]
SVEN [57.7K]
Assuming the question marks are minus signs
to find max, take derivitive and test 0's and endpoints

take derivitive
f'(x)=18x²-18x-108
it equal 0 at x=-2 and 3
if we make a sign chart to find the change of signs
the sign changes from (+) to (-) at x=-2 and from (-) to (+) at x=3
so a reletive max at x=-2 and a reletive min at x=3

test entpoints

f(-3)=83
f(-2)=134
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the min is at x=3 and max is at x=-2
4 0
3 years ago
2. In a given population of two-earner male-female couples, male earnings have a mean of $40,000 per year and a standard deviati
Kobotan [32]

Answer: $85,000

Step-by-step explanation:

Given : In a given population of two-earner male-female couples, male earnings have a mean of $40,000 per year and a standard deviation of $12,000.

\mu_M=40,000\ \ ;\sigma_M=12,000

Female earnings have a mean of $45,000 per year and a standard deviation of $18,000.

\mu_F=45,000\ \ ;\sigma_F=18,000

If  C denote the combined earnings for a randomly selected couple.

Then, the mean of C will be :-

\mu_c=\mu_M+\mu_F\\\\=40,000+45,000=85,000

Hence, the mean of C = $85,000

8 0
3 years ago
Suppose that $2000 is placed in a savings account at an annual rate of 4.6%, compounded quarterly. Assuming that no withdrawals
LenaWriter [7]

Answer:

It will take 35.46 quarters for the account to grow to $3000.

Step-by-step explanation:

Since the annual rate is compounded quarterly, this can be calculated using the formula for calculating the future value as follows:

FV = PV * (1 + r)^n ............................ (1)

Where;

FV = future value or the amount the deposit expected to grow to = $3,000

PV = Present value or the amount place in the savings = $2,000

r = Quarterly rate = Annual rate / 4 = 4.6% / 4 = 0.046 / 4 = 0.0115

n = number of quarters it will take for the loan to grow to $3000 = ?

Substituting the values into equation (1) and solve for n, we have:

$3,000 = $2,000 * (1 + 0.0115)^n

$3,000 / $2,000 = (1.0115)^n

1.50 = (1.0115)^n

Loglinearise both sides, we have:

log(1.50) = n log(1.0115)

0.176091259055681 = n * 0.00496588710682352

n = 0.176091259055681 / 0.00496588710682352

n = 35.4601816891322

Rounding to the nearest hundredth, which also implies to rounding to 2 decimal places, we have:

n = 35.46

Since the the annual rate is compounded quarterly, it will therefore take 35.46 quarters for the account to grow to $3000.

3 0
2 years ago
Asher opened a savings account by making an initial deposit and setting up weekly automatic deposits as shown in the graph. How
Sergio [31]

Answer:

  • y = 7.5x + 20

Step-by-step explanation:

As per the graph, the y-intercept is 20 which is the initial amount.

Another marked point has coordinates (4, 50).

<u>The slope is:</u>

  • m = (50 - 20)/4 = 7.5

<u>The equation of the line is:</u>

  • y = 7.5x + 20

$20 is the initial deposit and weekly deposit is $7.5

5 0
2 years ago
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