Section 4.1 Sets
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If A is a subset of B, we denote the relationship by A # B. If A # B but A ∙ B (there
is at least one element of B that is not an element of A), then A is a proper subset
of B, denoted by A ( B.
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PraCtiCe 5 Use formal logic notation to define A ( B.
example 3
Let
A = 51, 7, 9,156
B = 57, 96
C = 57, 9, 15, 206
Then the following statements (among others) are all true:
B # C
15 [ C
B # A
{7, 9} # B
B ( A
{7} ( A
A h C
[ # C
The last statement ([ # C  ) is true because the statement (4x)(x [ [ S x [ C  ) is
true because x [ [ is always false.
ReminDeR
Be sure you understand
the difference between
the symbols [ (element of)
and # (subset of).
Suppose that B = {x 0 P(x)} and that A # B. Because every element of A is
also an element of B, and P is a property characterizing all elements of B, then
every element in A also has property P(x). The elements of A “inherit” property
P. In fact, to prove that A # B, we pick an arbitrary x [ A and show that P(x)
PraCtiCe 6 Let
A = {x 0 x [ ℕ and x ≥ 5}
B = {10, 12, 16, 20}
C = {x 0 (E y)( y [ ℕ and x = 2y)}
Which of the following statements are true?
a. B # C
g. {12} [ B
b. B ( A
h. {12} # B
c. A # C
i. {x 0 x [ ℕ and x < 20} h B
d. 26 [ C
j. 5 # A
e. {11, 12, 13 } # A
k. {[} # B
f. {11, 12, 13} ( C
l. [ o A
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