Notes:
The notation ">=" without quotes means "greater than or equal to"
The upper case "U" means "set union"
Instead of using the intersection symbol, I will use a lower case 'n'
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Problem 1
A = {x | x < 1} which is the set of x values smaller than 1
B = {x | x >= 5} is the set of x values that are equal to 5 or larger
A U B = set of values that are from set A OR they are from set B (or both)
A U B = {x | x < 1 or x >= 5}
we simply connect the two inequalities mentioned with an "or"
note: how there is no overlap between the two regions. The "U" means "set union" which is like a sort of glue to tie the two sets together with an "or".
Answer: {x| x < 1 or x >= 5}
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Problem 2
A = {x | x < 1}
C = {x | x = 5} which is the set of one value only: 5 (x cannot equal any other value)
A U C = {x| x < 1 or x = 5}
So if a number is in set A U C, then this number is either less than 1, OR it is equal to 5
Answer: {x|x < 1 or x = 5}
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Problem 3
B U C = {x | x >= 5} because set C already has the "or equal to" part in there.
Set C is a subset of set B. If an item is in set C, then it is also in set B.
Answer: {x| x >= 5}
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Problem 4
Again recall that I'm using an 'n' to indicate "set intersection" instead of the upside down "U" symbol
A n B is the set of items that are in BOTH sets A and B at the same time. From problem 1, I mentioned there's a gap. There is no x value that is both less than 1 AND greater than or equal to 5. So this means that
A n B = empty set
which we use the "O" with a slash through it. This is a special symbol to indicate "empty set"
Another way to write "empty set" is to use curly braces with nothing inside like so { }
Answer: The "O" with a slash through it
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Problem 5
B n C is the same as {x| x = 5}
Why? Because if an item is in B n C, then it has to be in BOTH set B and set C at the same time. The only way this happens is if x = 5. If x is any other value, then it won't be in set C
Answer: {x| x = 5}
T=number of shirt
p=pants
s=shoes
t=24
8 fewer shoes than pants means s=p-8
t shirts are double pants, means t=2p
so
t=24
s=p-8
t=2p
solve
we can see
24=t=2p
24=2p
divide both sides by 2
12=p
sub into other equation
s=p-8
s=12-8
s=4
4 pairs of shoes
Answer:
38
Step-by-step explanation:
By CPCTC, angles C and D are congruent.
130 = 4x - 14
144 = 4x
x = 38
Answer:
- Determine the null and alternative hypotheses.
- Select a level of significance.
- Draw a random sample, and calculate the sample statistic.
- Decide whether to reject or fail to reject the null hypothesis.
Step-by-step explanation:
The steps that involve testing a hypothesis can be sorted like this:
First, we have to determine the null and alternative hypothesis. This is related directly to the claim we are trying to prove.
Second, we select a level of significance. This depends on how conservative we want to be with the conclusions. We have to do it before evaluating the test statistic, although it is not involved in its calculations.
Third, we draw a random sample, and calculate the sample statistic. This is the core of the hypothesis test, were we use the information of a sample to infer a characteristic of the population.
Fourth and last, with the results of the test statistic, we decide whether to reject or fail to reject the null hypothesis. This leads to the conclusions about the claims.
a.
is a proper joint density function if, over its support, 
is non-negative and the integral of is 1. The first condition is easily met as long as 
. To meet the second condition, we require
b. Find the marginal joint density of 
and 
by integrating the joint density with respect to 
:
Then
c. This probability can be found by simply integrating the joint density:
