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liubo4ka [24]
3 years ago
12

In each part, decide if the statement is true. If it is true, prove it. If it is not true, give an explicit counterexample and d

emonstrate that your counterexample really is one.
(a) For any sets A, B and C, A ∩ B ⊆ C if and only if either A ⊆ C or B ⊆ C.

(b) For any sets A, B and C, A ⊆ B ∩ C if and only if both A ⊆ B and A ⊆ C
Mathematics
1 answer:
svet-max [94.6K]3 years ago
4 0

Answer: Hello!

ok, remember that "if and only if" implies that you need to prove the statement in both ways, this is represented with the ⇔ usually.

a) For any sets A, B and C, A ∩ B ⊆ C if and only if either A ⊆ C or B ⊆ C.

In this type of problems, i  find very useful start looking for some counterexample.

In this case, suppose that A = {1,2,3,4,5} , B = {3,4,5,6,7} and C = {3,4,5,6}

then is easy to see that A ⊄ C and B ⊄C.

And A∩B = {3,4,5}

then A∩B ⊂ C

then the statement is false (because one of the ways is false, remember that this is an "if and only if" statement)

b) For any sets A, B and C, A ⊆ B ∩ C if and only if both A ⊆ B and A ⊆ C.

the first way is true; because if A ⊆ B ∩ C. then all the elements of A are in the intersection of B and C (which are common elements for B and C) and then all the elements of A are in the set B and in the set C, and this means that A ⊆ B and A ⊆ C.

But let's see the other way now, suppose that A ⊆ B and A ⊆ C, now we want to know if A ⊆ B ∩ C.

if A ⊆ B and A ⊆ C, means that all the elements of A are in B, and all the elements of A are in C, then all the elements of A are common elements between B and C, this means that B ∩ C is at least equal to A (at least, because we know that all the elements of A are common elements between B and C, but there could be more common elements that don belong to A)

then A ⊆ B ∩ C.

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