Problems from The principle of addition and multplication

Calculate the cardinal of $A \cup B$ and $A \times B$ if $A = {1, 3, 4, 5, 6, 7}$ and $B = {3, 4, 7, 8, 9}$ .

See development and solution

Development:

First the cardinal ones are calculated of $A$ and $B$ : $c a r d (A) = 6$ $c a r d (B) = 5$ Next the intersection of both sets is calculated: $A \cap B = {3, 4, 7}$ and later its cardinal one: $c a r d (A \cap B) = 3$ Finally, to calculate $c a r d (A \cup B)$ we apply the principle of addition, and to calculate the cardinal one of $A \times B$ we apply the principle of multiplication $c a r d (A \cup B) = c a r d (A) + c a r d (B) - c a r d (A \cap B) = 6 + 5 - 3 = 8$ $c a r d (A) \times c a r d (B) = 6 \cdot 5 = 30$

Solution:

$c a r d (A \cup B) = 8$ ; $c a r d (A) \times c a r d (B) = 30$

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