This document discusses mathematical sets and their properties including: - Elements and membership in sets - Identical sets, subsets, the null/empty set, and disjoint sets - Set operations including union, intersection, and complement - Necessary and sufficient conditions for implications between sets - Functions and their notation as mappings between sets It provides examples and references to textbook pages for further information on these set theory and function concepts.

1. shito@seinan-gu.ac.jp
2020 4 18
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•
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1 P.14
(1)
•
• or
• (element)
(is an element of)
(2)
• (identical set):
S1 = {2, 7, a, f} S2 = {f, 7, a, 2}
=⇒
S1
S2
1

2. www.seinan-gu.ac.jp/˜shito 2020 4 18 9:41
• (subset):
S = {1, 2, 3, 4}, T = {3, 4}
=⇒ (T is contained in S.)
=⇒ (S includes T.)
S
T
: x ∈ T x ∈ S T ⊂ S
=⇒ S ⊂ S
• (null set, empty set):
=⇒
otherwise
• (disjoint):
I = {1, 3}, J = {4, 5} I J
(3)
•
BA A ∪ B = {x | x ∈ A or x ∈ B}
A = {1, 2}, B = {2, 3}, A ∪ B = C =⇒ C =
(union)
•
BA A ∩ B = {x | x ∈ A and x ∈ B}
A ∩ B = C =⇒ C =
(intersection)
• (complement): ˜A
II 2

3. www.seinan-gu.ac.jp/˜shito 2020 4 18 9:41
AA
~
普遍集合 U (universal set)
2
(1) : x P A = {x | x }
B = {x | x }
(2) : A =⇒ B
A B
(If A then B.)
B
A
• B A (necessary condition)
• A B (sufficient condition) A only if B.
(3) (necessary and sufficient condition): A ⇐⇒ B
A B B A
A =⇒ B
A only if B
B =⇒ A
A if B
A if and only if B. (A iff B.)
⇓
or
A
B
A B
(4)
A B
B
~
B が成立するか
しないか２つに
１つしかない．
なぜなら
B + B = U
~こちらの道を断つ
II 3

4. www.seinan-gu.ac.jp/˜shito 2020 4 18 9:41
(5)
A =⇒ B ⇐⇒ ˜B =⇒ ˜A
• pp.14–21, pp.105–107
• P.21 2.3 1–4, 7
• A ⊂ B B ⊂ A A = B
3
? x y y x y = f(x)
x y
y x
x
y
x1 x2
y0
y1
y2
x0
x
y
4
y = f(x): x y
(1)
y = f(x) = 7 y x
II 4

5. www.seinan-gu.ac.jp/˜shito 2020 4 18 9:41
(2) 1
y = f(x) = 3 + 1
2 x
(1) pp.22–40
(2) 2.5 1, 2, 5–8
II 5