lets-two-sets-A-B-and-take-X-the-number-of-elements-of-the-set-X-them-proof-or-give-a-counter-example-that-if-A-B-and-A-B-then-A-and-B-

Question Number 1616 by 123456 last updated on 27/Aug/15

lets two sets A, B and take ∣ X ∣ the number

of elements of the set X, them

proof or give a counter example that

if ∣ A \cup B ∣= \infty and ∣ A \cap B ∣= \infty then ∣ A ∣= \infty and ∣ B ∣= \infty

Commented by 112358 last updated on 27/Aug/15

A \cup B = A + B - A \cap B

\Rightarrow∣ A \cup B ∣=∣ A ∣ + ∣ B ∣ - ∣ A \cap B ∣

∴ ∣ A \cap B ∣ + ∣ A \cup B ∣=∣ A ∣ + ∣ B ∣

B y l o g i c, p \to q \equiv∽ q \to∽ p . S o,

p r o p o s e t h e f o l l o w i n g s t a t e m e n t s

p :∣ A \cap B ∣ a n d ∣ A \cup B ∣ a r e n o n - f i n i t e .

q : ∣ A ∣ a n d ∣ B ∣ a r e n o n - f i n i t e .

T h e n o n e m a y f i n d i t e a s i e r t o

s h o w t h a t ∽ q \to∽ p r a t h e r t h a n

p \to q d i r e c t l y .

L e t ∣ A ∣ a n d ∣ B ∣ b e f i n i t e .

\Rightarrow ∣ A ∣= n, ∣ B ∣= m

w h e r e n, m \in Z^{+} a n d n, m a r e f i n i t e .

\Rightarrow∣ A \cup B ∣ + ∣ A \cap B ∣= m + n

∵ n a n d m a r e f i n i t e

\Rightarrow m + n i s f i n i t e

\Rightarrow∣ A \cap B ∣ + ∣ A \cup B ∣ i s f i n i t e

∴ ∣ A \cap B ∣= x, ∣ A \cup B ∣= y s o t h a t

x + y = m + n

w i t h x, y \in Z^{+} a n d a r e f i n i t e . T h u s,

∣ A \cup B ∣ a n d ∣ A \cap B ∣ a r e f i n i t e .

∵ ∽ q \to∽ p t h e n w e h a v e t h a t

p \to q . H e n c e, ∣ A \cap B ∣= \infty a n d ∣ A \cup B ∣= \infty

i m p l i e s t h a t ∣ A ∣= \infty a n d ∣ B ∣= \infty .

Commented by 123456 last updated on 28/Aug/15

nice : D

thanks

Commented by Rasheed Ahmad last updated on 28/Aug/15

A p p r e c i a t i o n s! G o o d a p p r o a c h!