I) Prove that every function can be represented as a sum of an even function and an odd function.
II)Let A, B, C be subsets of a set S. Prove the following statements and illustrate them with Venn Diagrams:
2a) The famous DeMorgan’s laws in their basic forms: and
. Assume that both sets A and B are subsets of Set S. In words, the first is: union of complements is the complement of intersection; the second is: intersection of two complements is the complement of the union of the two sets.
Sample Solution:
Let us say that we need to prove: .
Proof: It must be shown that the two sets have the same elements; in other words, that each element of the set on LHS is an element of the set on RHS and vice-versa.
If , then
and
. This means that
, and
and
. Since
and
, hence
. Hence,
.
Conversely, if , then
and
. Therefore,
and
. Thus,
and
, so that
. QED.
2b) .
2c)
III) Prove that if I and S are sets and if for each , we have
, then
.
Sample Solution:
It must be shown that each element of the set on the LHS is an element of the set on RHS, and vice-versa.
If , then
and
. Therefore,
, for at least one
. Thus,
, so that
.
Conversely, if , then for some
, we have
. Thus,
and
. Since
, we have
. Therefore,
. QED.
IV) If A, B and C are sets, show that :
4i)
4ii)
4iii)
4iv)
V) Let I be a nonempty set and for each let
be a set. Prove that
5a) for any set B, we have :
5b) if each is a subset of a given set S, then
VI) Prove that if ,
, and
are functions, then :
VII) Let be a function, let A and B be subsets of X, and let C and D be subsets of Y. Prove that:
7i) ; in words, image of union of two sets is the union of two images;
7ii) ; in words, image of intersection of two sets is a subset of the intersection of the two images;
7iii) ; in words, the inverse image of the union of two sets is the union of the images of the two sets.
7iv) ; in words, the inverse image of intersection of two sets is intersection of the two inverse images.
7v) ; in words, the inverse of the image of a set contains the set itself.
7vi) ; in words, the image of an inverse image of a set is a subset of that set.
For questions 8 and 9, we can assume that the function f is and a set A lies in domain X and a set C lies in co-domain Y.
8) Prove that a function f is 1-1 if and only if for all
; in words, a function sends different inputs to different outputs iff a set in its domain is the same as the inverse of the image of that set itself.
9) Prove that a function f is onto if and only if for all
; in words, the image of a domain is equal to whole co-domain (which is same as range) iff a set in its domain is the same as the image of the inverse image of that set.
Cheers,
Nalin Pithwa