[imath]\bold{Challenge}[/imath] - 2

[logistic_guy]

Set \(\displaystyle A\) consists of all multiples of \(\displaystyle 5\) greater than \(\displaystyle 10\) and less than \(\displaystyle 100\). Set \(\displaystyle B\) consists of all multiples of \(\displaystyle 8\) greater than \(\displaystyle 16\) and less than \(\displaystyle 100\). Show that each conjecture is false by finding a counterexample.

\(\displaystyle \bold{a.}\) Any number in set \(\displaystyle A\) is also in set \(\displaystyle B\).
\(\displaystyle \bold{b.}\) Any number less than \(\displaystyle 100\) is either in set \(\displaystyle A\) or in set \(\displaystyle B\).
\(\displaystyle \bold{c.}\) No number is in both set \(\displaystyle A\) and set \(\displaystyle B\).

 

[logistic_guy]

\(\displaystyle A = \bigg\{15,20,25,30,35,40,45,50,55,60,65,70,75,80,85,90,95\bigg\}\)


\(\displaystyle B = \bigg\{24,32,40,48,56,64,72,80,88,96\bigg\}\)

 

[khansaheb]

2^n * 5 * 8 → 40, 80

 

[logistic_guy]

\(\displaystyle \bold{a.}\) Any number in set \(\displaystyle A\) is also in set \(\displaystyle B\).

This conjecture is false because \(\displaystyle 15 \in A\) but \(\displaystyle 15 \notin B\).

 

[logistic_guy]

\(\displaystyle \bold{b.}\) Any number less than \(\displaystyle 100\) is either in set \(\displaystyle A\) or in set \(\displaystyle B\).

This conjecture is false because \(\displaystyle 99 \notin A\) and \(\displaystyle 99 \notin B\) as well.

 

[logistic_guy]

\(\displaystyle \bold{c.}\) No number is in both set \(\displaystyle A\) and set \(\displaystyle B\).

This conjecture is false because \(\displaystyle 40 \in A\) and \(\displaystyle 40 \in B\) as well.

 

[khansaheb]

c. No number is in both set A\displaystyle AA and set B\displaystyle BB.

40 and 80 are in both set A and set B