True and False?

If the product $A\times B$ of two sets $A$ and $B$ is the empty set , then both $A$ and $B$ have to be empty set.

If the product $A\times B$ of two sets $A$ and $B$ is the empty set , then both $A$ and $B$ have to be empty set.

posader86
2022-07-20
Answered

True and False?

If the product $A\times B$ of two sets $A$ and $B$ is the empty set , then both $A$ and $B$ have to be empty set.

If the product $A\times B$ of two sets $A$ and $B$ is the empty set , then both $A$ and $B$ have to be empty set.

You can still ask an expert for help

Octavio Barr

Answered 2022-07-21
Author has **11** answers

False, one of them being empty suffices. This one is at the heart of possible confusion about $k\times l$ matrices with $k=0$ or $l=0$ (possibly both). You need to distinguish various types of empty matrices if you want to define multiplication correctly (a $n\times 0$ matrix multiplied by a $0\times m$ matrix gives a non-empty, though zero, $n\times m$ matrix), but you cannot make this distinction if you define matrices as a map on $[k]\times [l]$ assigning entries to positions, because $[k]\times [0]=\mathrm{\varnothing}=[0]\times [l]$.

asked 2022-05-08

My problem is the following. I like to know if there exist a sentence true in complex a field but false in a field of positive characteristic.

asked 2022-04-07

The prevalence of breast cancer in women over 40 in country X is estimated to be 0.8% (i.e., 8 in every 1,000 women in that age group).

Mammograms test for the presence of breast cancer. A positive result indicates that the disease is present. A negative result indicates that it is not.

The sensitivity of a mammogram test for breast cancer is estimated to be 90%. This is the probability that the mammogram will give a positive result when the person being tested does have breast cancer.

The false positive rate for the mammogram is 7.5%. This is the probability that the mammogram will give a positive test result when the person being tested does not have breast cancer.

All women who test positive (816) in the mammogram are referred for a further, different examination, which however has the same sensitivity and false positive rates as the first test.

What is the probability that a woman referred for this examination and testing positive again, actually does have breast cancer?

Mammograms test for the presence of breast cancer. A positive result indicates that the disease is present. A negative result indicates that it is not.

The sensitivity of a mammogram test for breast cancer is estimated to be 90%. This is the probability that the mammogram will give a positive result when the person being tested does have breast cancer.

The false positive rate for the mammogram is 7.5%. This is the probability that the mammogram will give a positive test result when the person being tested does not have breast cancer.

All women who test positive (816) in the mammogram are referred for a further, different examination, which however has the same sensitivity and false positive rates as the first test.

What is the probability that a woman referred for this examination and testing positive again, actually does have breast cancer?

asked 2022-04-30

3% of the population has disease X.

A laboratory blood test has

(a) 96% effective at detecting disease X, given that the person actually has it.

(b) 1% “false positive” rate. i.e, a person who does not have disease X has a probability of 0.01 of obtaining a test result implying they have the disease.

What is the probability a person has the disease given that the test result is positive?

A laboratory blood test has

(a) 96% effective at detecting disease X, given that the person actually has it.

(b) 1% “false positive” rate. i.e, a person who does not have disease X has a probability of 0.01 of obtaining a test result implying they have the disease.

What is the probability a person has the disease given that the test result is positive?

asked 2022-06-13

A town with a population 10,000 has suffered an outbreak of dragon pox, with 3% of the population being infected. There is a test to diagnose the disease. If you have the dragon pox, the test will correctly register positive 97% of the time and will yield a false negative 3% of the time. If you do not have the dragon pox the test will register that correctly 97% of the time and will yield a false positive 3% of the time. Given that you tested positive for the disease, what is the probability that you actually have dragon pox?

asked 2022-07-18

True and False?

There is no one-to-one correspondence between the set of all positive integers and the set of all odd positive integers because the second set is a proper subset of the first.

There is no one-to-one correspondence between the set of all positive integers and the set of all odd positive integers because the second set is a proper subset of the first.

asked 2022-04-06

What is the rationale behind ROC curves?

I am not sure how ROC curves work. I see that the X-Axis is the false positive rate while the Y axis is the true positive rate.

1) I don't understand how for a given statistical learning model, you could have the true positive and false positive rate to vary from 0 to

1. Are you changing parameters in the model to make it so?

2) What about true negatives and false negatives? How are these represented in the curve?

I am not sure how ROC curves work. I see that the X-Axis is the false positive rate while the Y axis is the true positive rate.

1) I don't understand how for a given statistical learning model, you could have the true positive and false positive rate to vary from 0 to

1. Are you changing parameters in the model to make it so?

2) What about true negatives and false negatives? How are these represented in the curve?

asked 2022-06-29

A cancer test is 90 percent positive when cancer is present. It gives a false positive in 10 percent of the tests when the cancer is not present. If 2 percent of the population has this cancer what is the probability that someone has cancer given that the test is positive?

I multiplied the 90 by 10 divided by 90 times 10 plus 2.

I multiplied the 90 by 10 divided by 90 times 10 plus 2.