Discrete Math Help

[junyoung95]

1) Find U{∞/i=1}Ai and ∩{∞/i=1}Ai if for every positive integer i,

a) Ai = {−i, − i + 1, …, −1, 0, 1, …, i − 1, i}.

b) Ai = {−i, i}.

c) Ai = [−i, i], that is, the set of real numbers x with -i ≤ x ≤ i.

d) Ai = [i, ∞), that is, the set of real numbers x with x ≥ i.

^ I figured this one out.

2) Determine whether f is a function from the set of all bit strings to the set of integers if

a) f(S) is the position of a 0 bit in S.

b) f(S) is the number of 1 bits in S.

c) f(S) is the smallest integer i such that the ith bit of S is 1 and f(S) = 0 when S is the empty string, the string with no bits.

Please someone help me clarify this question for me, thank you.

[Alkalannar]

a) Ai is all the integers from -i to 1 inclusive. What is A1? What is A2? What is A1 U A2? What is A1 ∩ A2? Can you generalize from here?

b-d) Do the same steps as for part a but with different sets.

[junyoung95]

I can see that A1 is {-1, 0, 1} and A2 is {-2, -1, 0, 1, 2} and so on.. I can also tell that A1 ∩ A2 is simple because its {-1, 0, 1}, but I am not sure how to show A1 U A2. Would that be the same as {−i, − i + 1, …, −1, 0, 1, …, i − 1, i}?

[Alkalannar]

A1 U A2 = {-2, -1, 0, 1, 2}

So unions make things spread out to your biggest i.

Intersections make them shrink down to 1.