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3 years ago

Set A and the universal set U are defined as follows.

Mathematics

1 answer:

siniylev [52]3 years ago

5 0

Part (a)

<h3>Answer: Ø</h3>

This is the empty set

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Explanation:

It doesn't matter what set A is composed of. Intersecting any set with the empty set Ø will always result in the empty set.

This is because we're asking the question: "What does some set A and the empty set have in common?". The answer of course being "nothing" because there's nothing in Ø. Not even the value zero is in this set.

We can write Ø as { } which is a set of curly braces with nothing inside it.

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Part (b)

<h3>Answer: {1,2,3,4,5,6}</h3>

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Explanation:

When you union the universal set with any other set, you'll get the universal set.

The rule is $A \cup B = B$ where I've made B the universal set to avoid confusion of the letter U and the union symbol $\cup$ which looks nearly identical.

Why does this rule work? Well if an item is in set $\overline{A}$ , then it's automatically in set U (everything is in set U; it's the universe). So we're not adding anything to the universe when applying a union involving this largest set.

It's like saying

A = set of stuff inside a persons house
$\overline{A}$ = set of stuff outside a persons house (ie stuff that is not in set A)
U = set of every item

we can see that $\overline{A} \cup U$ will basically form the set of every item, aka the universal set.

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y = A.sin(B.t - C) + D or y = Acos(B.t - C) + D

where:

A is amplitude A=|A|

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C is the phase shift or horizontal shift: $\frac{C}{B}$

D is the vertical shift

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<u>Amplitude</u>:

a = $\frac{largest - smallest}{2}$

At t=0, Pluto is the farthest from the sun, a distance 6.9 billions km away. At t=66, it is closest to the star, P(66) = 4.4 billions km. Then:

a = $\frac{6.9-4.4}{2}$

a = 1.25

A time period for Pluto is T=66 years:

66 = $\frac{2.\pi}{b}$

b = $\frac{\pi}{33}$

<u>Vertical</u> <u>Shift</u>

It can be calculated as:

d = $\frac{largest+smallest}{2}$

d = $\frac{6.9+4.4}{2}$

d = 5.65

Knowing a, b and d, substitute in the equivalent positions and find P(t).

P(t) = a.sin(b.t) + d

P(t) = 1.25.sin( $\frac{\pi}{3}$ .t) + 5.65

The Pluto's distance from the sun as a function of time is

P(t) = 1.25.sin( $\frac{\pi}{3}$ .t) + 5.65

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3 years ago

Read 2 more answers

For an integer x,x squared minus x will always produce an even value

yuradex [85]

Answer:

Step-by-step explanation:

Not exactly sure what your question is - I am assuming that it is something like:

Show/prove that for any integer x, x^2 - x is even.

Suppose that x is an even integer. The product of an even integer and any other integer is always even (x = 2n, so x * y = 2 n * y which is even. Therefore x^2 is even. An even minus an even is even. (The definition of an even number is that it is divisible by 2 or has a factor of 2. So the difference of even numbers could be written as 2*( the difference of the two numbers divided by 2); therefore the difference is even)

Suppose that x is an odd integer. The product of 2 odd numbers is odd - each odd number can be written as the sum of an even number and 1; multiplying the even parts with each other and 1 will produce even; multiplying the 1's will produce 1, so the product can be written as the sum of an even number and 1 - which is an odd number. The difference between two odd numbers is even - the difference between the even parts is even (argument above), the difference between 1 and 1 is zero, so the result of the difference is even.

x^2 is therefore even if x is even and odd if x is odd; The difference x^2 - x is even by the arguments above.

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