I need help with this problem:
If N= {1,2,3,4,....}, A= {x/x=2n-1, n E N}, B={x/x=2n, n E N}, and C= {x/x+2n +1, n=0, or n E N}, find the simplest possible expression for each of the following:
a. A U C
b. A U B - the U stands for union
c. A (intersects) I can't find the figure) C
d. A (intersects) B
Also, I used the E as ELEMENT in the top part of the question. Hopefully it isn't too confusing! I am very lost on this problem!
It might help if you gave A,B and C names so you can see what the sets represent. Write out the first 3 or 4 terms if you're not sure, but you should learn to identify properties too.
For A the first three terms are 1,3,5, the first three terms of B are 2,4,6 and the first three are 1,3,5.
You should be able to see that A is the positive odd numbers, B is the positive even numbers and C is also the positive odd numbers.
Since A and C are the same sets, what is the union of them?
Since A and B are the natural separation of the positive integers into odd and even numbers, what is their union?
What is the intersection of a set with itself? You can use int for intersection or just spell it out, I can see what you mean either way
Do A and B have any elements in common?
5t+2=7
You should post a new question when your question is not related to the first one in the thread. That way we won't miss your question.
You posted;
5t+2=7
I'm not sure what the question is, find t?
Subtract 2 from both sides to get
5t=5 then divide both sides by 5.
5t+2=7
5t+2=7
5t=5
t=1
Nice job! You've solved the equation correctly. The value of t is indeed 1. Keep up the good work!
To find the simplest possible expression for each of the sets:
a. A U C (A union C):
Since A and C have the same elements, the union of A and C would simply be A itself.
Therefore, A U C = A = {x/x = 2n-1, n E N}
b. A U B (A union B):
The union of A and B would include all the elements in A and all the elements in B.
A = {x/x = 2n-1, n E N} = {1, 3, 5, 7, ...}
B = {x/x = 2n, n E N} = {2, 4, 6, 8, ...}
So, A U B = {1, 2, 3, 4, 5, 6, 7, 8, ...}
c. A ∩ C (A intersects C):
The intersection of A and C would include the elements that are common to both sets.
A = {x/x = 2n-1, n E N} = {1, 3, 5, 7, ...}
C = {x/x = 2n+1, n = 0, or n E N} = {1, 3, 5, 7, ...}
Since A and C are the same sets, the intersection of A and C would be A itself.
Therefore, A ∩ C = A = {x/x = 2n-1, n E N}
d. A ∩ B (A intersects B):
The intersection of A and B would include the elements that are common to both sets.
A = {x/x = 2n-1, n E N} = {1, 3, 5, 7, ...}
B = {x/x = 2n, n E N} = {2, 4, 6, 8, ...}
Since A and B have no elements in common (odd numbers and even numbers do not overlap), the intersection of A and B would be the empty set (∅).
Therefore, A ∩ B = ∅.
Sure, let's break down the problem step by step.
First, let's analyze the sets A, B, and C:
- Set A: It is defined as the set of numbers where x = 2n-1, and n is an element of the set of natural numbers (N). In simpler terms, it represents the set of positive odd numbers.
- Set B: It is defined as the set of numbers where x = 2n, and n is an element of the set of natural numbers (N). In simpler terms, it represents the set of positive even numbers.
- Set C: It is defined as the set of numbers where x = n+2n+1, and n can be either zero or an element of the set of natural numbers (N). In simpler terms, it represents the set of positive odd numbers as well, just like A.
Now, let's find the simplest possible expression for each part of the question.
a. A U C (A union C):
Since A and C are both sets of positive odd numbers, their union will simply be the set of positive odd numbers. So the simplest possible expression for A U C is just A itself.
b. A U B (A union B):
The union of A and B represents all the positive odd numbers and positive even numbers. So the simplest possible expression for A U B is just the set of all positive integers (N).
c. A ∩ C (A intersects C):
The intersection of two sets represents the common elements between them. In this case, since A and C are the same sets (both representing the positive odd numbers), the intersection of A and C will be A itself.
d. A ∩ B (A intersects B):
To find the intersection of two sets, we need to find the common elements between them. In this case, A represents the positive odd numbers and B represents the positive even numbers. They have no common elements since odd numbers and even numbers are mutually exclusive. Therefore, the intersection of A and B will be an empty set ({}).
I hope this explanation helps you understand the problem and find the simplest possible expressions for each part! Let me know if you have any further questions.