F and H are sets of real numbers defined as follows.

Question

F= {z│z ≤ 3}

H= {z│z > 6}

Write F U H and F ∩ H using interval notation. If the set is empty, write Ø

Answer 1

F is the set of all real numbers that are less than or equal to 3. H is the set of real numbers greater than 6.

F U H is the union (or combination) of the two sets, so it is the set of all real numbers that are less than equal to 3 or greater than 6:

F U H = {z | z≤3 U z>6}

F ∩ H is the set of real numbers that are in both sets; but since F is less than or equal to 3 and H is greater than 6, there are no numbers that are in both sets. Hence the answer is the empty set, Ø:

F ∩ H = Ø

Answer 2

Hi Maria, F U H is the union of the two sets and consists of values of z in either or both of F and H, so F U H = {z | z ≤ 3 or z > 6}. F ∩ H is the intersection of the two sets and contains elements that are in both of F and H. Does the intersection contain any elements?

Answer 3

Hello Maria,

Given any two sets A and B, the union of A and B (denoted A U B) is the set of all elements that are in A or B (or both). (Basically, the union operation just combines everything in A or B into one big set.) The intersection of A and B (denoted A ∩ B) is the set of all elements that are in A and B. (That is, the intersection consists of those elements that are common to both A and B.) In your problem, the set F is the set of all real numbers to the left of (and including) 3, while the set H is the set of all real numbers to the right of 6. If you are familiar with interval notation, these sets can also be represented as F = (-∞,3] and

G = (6,∞). You would probably find it helpful to graph these intervals on the number line. Now from the definition of union given above, F U H is the set of real numbers in F or H (or both). If you look at a picture of these two sets on the number line, you will realize there is no shorthand way to represent the set. We can write the result as

F U H = {z|z≤3} U {z|z> 6}

F U H = {z|z≤3 or z> 6}

F U H = (-∞,3] U (6,∞)

Either of the last two forms could be used, depending on whether you want set-builder notation or interval notation. On the other hand, F ∩ H is the set of all real numbers that are in common to the two sets. Geometrically, the intersection represents the overlap of the two intervals (-∞,3] and (6,∞). It is clear that there is nothing in common to these two intervals, so the intersection is the empty set. That is,

F ∩ H = ∅.

Hope that helps!

William