Let A and B be sets. Show that the following are equivalent. (i) A n B = A (ii)
ID: 3076381 • Letter: L
Question
Let A and B be sets. Show that the following are equivalent.(i) A n B = A
(ii) A u B = B
(iii) A c B
Explanation / Answer
It suffices to show that (i) => (ii) => (iii) => (i) [(i) => (ii)] Assume x is in A u B. Then x is in A or x is in B. Since A = A n B, then in both cases x is in B. Therefore (A u B) c B. On the other hand, assume x is in B. Then x is in A or x is in B, that is, x is in A u B. Therefore B c (A u B). Hence A u B = B. [(ii) => (iii)] Assume x is in A. Then x is in A or x is in B, that is, x is in A u B. But A u B = B, so x is in B. Therefore A c B. [(iii) => (i)] Assume x is in A n B. Then x is in A, hence (A n B) c A. On the other hand, assume x is in A. But since A c B, then x is also in B. Hence x is in A n B, thus A c (A n B). Therefore A n B = A. QED :)
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