6 1. Numbers and Logic
Such a n implicatio n i s logicall y distinc t fro m it s converse : B implie s A.
I f x i s a William s student , the n x i s a huma n being . True .
I f x i s not a huma n being , the n x i s no t a William s student . Con -
trapositive, true .
I f x i s a human being , the n x i s a Williams student . Converse , fals e
in thi s case .
If i t happen s tha t th e implicatio n an d it s convers e ar e bot h true , i.e. , i f
A = B an d B =$ A , the n w e sa y tha t A an d B ar e logicall y equivalen t o
that
A i f an d onl y i f B (Ao B).
Example. Le t S c Z . Conside r th e statement s
A: Al l element s o f S ar e even . (Fo r al l x G S, x i s even. )
B: Som e elemen t o f S i s even . (Ther e exist s x G S, suc h tha t x i s
even.)
Does B = A? Certainl y no t i n general .
Does A = B? No t i n genera l becaus e i f S i s the empt y se t 0 , the n A i s
true (vacuously) , whil e B i s false !
1.9. Abbreviations . Understan d bu t avoi d abbreviation s suc h a s
A and V or ^ no t 3 exist s V for all .
1.10. Sets . A set A i s completely determine d b y the element s i t contains :
A = {x: x G A}.
If al l element s o f A ar e element s o f J3 , w e sa y tha t A i s a subset o f B
(A C B) o r tha t B contains A (B D A). Yo u hav e t o b e careful , becaus e
the word "contains " i s used both fo r subset s an d fo r elements , a s determine d
by context . Prope r containmen t (A C B an d A ^ B) i s denote d b y A £ B.
Some text s denot e ordinar y containmen t b y A C B t o remin d yo u tha t
equality is allowed. W e say that set s A, B intersect i f AnB ^ 0 . Th e symbo l
f)™=i An denote s the set of elements common to all of the sets A\, A2, A 3 , . . . :
00
x G ( l A
n
i f an d onl y i f x G An fo r al l n.
7 1 = 1
We defin e th e complemen t o f a se t
AC
= {x: x £ A},
and th e differenc e o f two set s
A- B = {x eA: B}.
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