1. let A and B be sets. Show that
A U (B - A)=A U B
2. determine whether f is a function from Z to R if
a) f(n)= +n
b) 1/(n square -4)
For 1. B-A is the same as B intersect ~A (That's the complement of A)
So A U B = A U (B int ~A) = (A U B) int (A U ~A) A U ~A is the universal set.
For 2. a) f(n) is the absolute value function, so yes, it's a function.
For 2. b) 1/(n^2 - 4) f is not defined at +/- 2
To show that A U (B - A) = A U B for sets A and B, we'll break it down step by step:
1. Start with the left side: A U (B - A)
This means we need to find the elements that are in A or in (B - A).
2. (B - A) represents the elements that are in set B but not in set A.
To find (B - A), we need to take all the elements in set B and remove the ones that are also in set A.
3. So, (B - A) can be written as B intersection A' (where A' is the complement of A).
The complement of A represents all the elements that are not in set A.
4. Now, the left side becomes A U (B intersection A').
This means we need to find the elements that are in set A or in the intersection of sets B and A'.
5. Using the distributive property of set operations, we can expand A U (B intersection A') as (A U B) intersection (A U A').
The union of A with B represents all the elements that are in either set A or set B.
The union of A with A' represents the universal set, as anything that is not in A will be part of A'.
6. Finally, (A U B) intersection (A U A') simplifies to (A U B) intersection U, which is just equal to A U B.
Therefore, we have shown that A U (B - A) = A U B.
For the second question, determining whether f is a function from Z (integers) to R (real numbers):
a) f(n) = +n
In this case, f(n) represents the positive value of n. For example, if n = -3, f(n) = 3.
Since every integer input n has a unique positive output, f is indeed a function from Z to R.
b) f(n) = 1 / (n^2 - 4)
In this case, we need to consider the domain of the function to determine if it is a function from Z to R.
The function is undefined when the denominator (n^2 - 4) equals zero, as division by zero is not defined.
Solving n^2 - 4 = 0 gives us n = -2 and n = 2.
This means the function is not defined for n = -2 and n = 2, as these values would result in division by zero.
Thus, we can conclude that f is not a function from Z to R.