What is the cardinality of the set {ø, {ø}}?

) zero 
b) one 
c) two (the members are the empty set and a set with one element (which happens to be the empty set)) 
d) three (the members are the empty set, a set containing the empty set and a set with two elements (those being the empty set and a set containing the empty set)) 

It takes some practice to readily sort out all the commas and brackets. Try drawing circles around the members.

a) ø 0
b) {ø} 1
c) {ø,{ø}} 2
d) {ø,{ø}, {ø,{ø}}} 3

#3. a) Section 1.6, page 85, #16: Can you conclude that A = B if A and B are 2 sets with the same power set? Why or why not?

The union of all the sets in the power set of X is X, so we can recover a set from its power set. The answer is “yes”

​b) Section 1.6, page 86, #22: Suppose that A x B = ø, where A and B are sets. What can you conclude?

One of A or B (or both) must be empty (if neither A nor B were empty, there would be an element in AxB)

#4. a) Section 1.7, page 95, #14a,e: Let A, B and C be sets. Show that
a) ( A U B) ( A U B U C)

(i) in words by showing the appropriate subset relations as done in class

Suppose x A U B
Then x A or B
Therefore x A U B U C
(truthfully, this is almost given to be true by the definition of union)

b) (B – A) U (C – A) = (B U C) – A

We need to show:

1. (B – A) U (C – A) Í (B U C) – A
and 2. (B U C) – A Í (B – A) U (C – A)

The equation of the circle is (x-1)2 + (y+6)2 = r2.

Step-by-step explanation:

The radius is the distance between (1,-6) and (6,-2)which can be found using the distance formula; call that number r.