Question

esponse Let A-xx is a natural number less than 6) and B=(xxis an even number from 2 to 101 What is the cardinality of AUB?*​

Answer 1

${\orange} \bf Question$

Let the above question be represented as :-

$\tt \: A = (x : x \epsilon \: N \: ; \: x<6 )$

$\tt \: B = ( x : x \: \epsilon \: 2n ; \: 1 \leqslant n \leqslant 50 )$

$\tt \: To \: Find \dots$

Cardinality of AUB i.e to find the total number of elements present in union of set A and B .

Solution

Let's find the set in Roster form first .

$\tt \: A = (1,2,3,4,5) \rightarrow n(A) = 5$

$\tt B = (2,4,6,8,10,12,14,...,100) \\ \ \tt\rightarrow n(B) = 50$

We have to find the set AUB and then only we can find its cardinality .

☯︎ Remember that

No element in a set should be written more than once i.e no repetition of elements.

We can observe that 2 and 4 are the only such elements showing repetition in both the set.

So , they must be counted once .

$\tt \: n(A \cap B) = 2$

We can now apply formula.

$\tt \: n(A \cup B) = n(A) + n(B) - n(A \cap B)$

On application of the values .

$\tt \: n(A \cup B) = 5 + 50 - 2 \\ = 55 - 2 = \colorbox{pink}{53}$

Conclusion

The cardinality of AUB = 53

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Additional Information

$\tt \: n(A - B) = n(A) - n(A \cap B)$

$\tt n(B - A ) = n(B) - n(A \cap B )$

$\tt \: n(A \cup B) = n(A - B) + n(B - A) + n(A \cap B)$

Thankyou

Answer 2

$\tt Answer$

$\bf \: Cardinality \: of \: AUB = \red{53}$

Hope it help you !!