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∣=∞


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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

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