epdf.pub_the-nuts-and-bolts-of-proofs-third-edition-an-intr

Special Kinds of Theorems 77
3. Prove the second of De Morgan's laws:
(A U By = A'n B'.
4. For any three sets A, B, and C, the following equaUty holds:
{AC\B)C\C =
Ar\{Br\C)
(associative property of intersection)
5. Prove or disprove the following statement:
The sets ^ = {all integer multiples of 16 and 36} and 5 = {all integer
multiples of 576} are equal.
6. Prove or disprove the following equalities, where A, B, and C are
subsets of universal set U:
7. The sets:
a. AU(BnC) = (AUB)nC;
b. (ADBn Cy = AUB'U C\
A = {(x,y)\y =: x^ - 1 with x G D? and x G K}
and
are equal.
f x^- 1
B = Ux, y)\y = —^—7 with x G IR and y e
8. Prove by induction that (^i n .42 Pi... n ^„)'= ^; U ^'2 ^ ... U A^ for
all n > 2.
(See Example 5 for the base case n = 2.)
9. Prove by induction that, if a set has n elements, then it has 2" subsets,
where ^i > 0.
Fill in the details in the following proof.
10. A set ^c[R is convex if, whenever x and y are elements of A, the
number tx-h{l- t)y is an element of A for all values of t with 0<t<l.
The set {z|z = tx + (1 - 0)^ for 0 < t < 1} is called the line segment
joining x and y.
Empty sets (sets with zero elements) and sets with one element are
assumed to be convex.
Given this information, outhne the proof of the following statement:
The intersection of two or more convex sets is a convex set.