Determine whether the following statements are true or false. Justify your answer.
"SUFFICIENT conditions GUARANTEE!"
"NECESSARY conditions GET GUARANTEED!"
i) A sufficient condition for an integer to be divisible by 8 is that it is
divisible by 16.
True, it's a SUFFICIENT condition, because it's ENOUGH information to know that
an integer is divisible by 16, to be sure that it's divisible by 8.
[It's a SUFFICIENT condition to know that 48 is divisible by 16, for that's
ENOUGH information to tell us that 48 is divisible by 8. However, it's not a
NECESSARY condition because, for instance, 24 is NOT divisible by 16, yet 24
is divisible by 8.]
ii) A necessary condition for an integer to be divisible by 6 is that it be
divisible by 2.
True, it's a NECESSARY condition, because we MUST KNOW that an integer is
divisible by 2, i.e., that it is even, in order for it to be divisible by 6.
[It's a NECESSARY condition to know that 42 is divisible by 2, for it to be
divisible by 6. However, it's not a SUFFICIENT because 22 is divisible by 2
but 22 is not divisible by 6.]
For all integers a,b, and c, if a|(b + c), then a|b or a|c
That's false because 3|(7+8) but it is not true because 3|7 and 3|8 are both
false.
[However "a|b and a|c" is a sufficient (but not necessary) condition that
a|(b+c).]
Edwin