calculus

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: $A^{'} \bigcup B^{'} = (A \bigcap B)^{'}$ and $A^{'} \bigcap B^{'} = (A \bigcup B)^{'}$ . 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: $A^{'}\bigcap B^{'}=(A \bigcup B)^{'}$ .

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 $x \in A^{'} \bigcap B^{'}$ , then $x \in A^{'}$ and $x \in B^{'}$ . This means that $x \in S$ , and $x \notin A$ and $x \notin B$ . Since $x \notin A$ and $x \notin B$ , hence $x \notin A \bigcup B$ . Hence, $x \in (A \bigcup B)^{'}$ .

Conversely, if $x \in (A \bigcup B)^{'}$ , then $x \in S$ and $x \notin A \bigcup B$ . Therefore, $x \notin A$ and $x \notin B$ . Thus, $x \in A^{'}$ and $x \in Y^{'}$ , so that $x \in A^{'} \bigcap B^{'}$ . QED.

2b) $A \bigcap (B \bigcup C) = (A \bigcap B)\bigcup (A \bigcap C)$ .

2c) $A \bigcup (B \bigcap C) = (A \bigcup B) \bigcap (A \bigcup C)$

III) Prove that if I and S are sets and if for each $i \in I$ , we have $X_{i} \subset S$ , then $(\bigcap_{i \in I} X_{i})^{'} = \bigcup_{i \in I}(X_{i})^{'}$ .

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 $x \in (\bigcap_{i \in I} X_{i})^{'}$ , then $x \in S$ and $x \notin \bigcap_{i \in I} X_{i}$ . Therefore, $x \notin X_{i}$ , for at least one $j \in I$ . Thus, $x \in (X_{i})^{'}$ , so that $x \in \bigcup_{i \in I}(X_{i})^{'}$ .

Conversely, if $x \in \bigcup_{i \in I}(X_{i})^{i}$ , then for some $j \in I$ , we have $x \in (X_{i})^{'}$ . Thus, $x \in S$ and $x \notin X_{i}$ . Since $x \notin X_{i}$ , we have $x \notin \bigcap_{i \in I}X_{i}$ . Therefore, $x \in \bigcap_{i \in I}(X_{i})^{'}$ . QED.

IV) If A, B and C are sets, show that :

4i) $(A-B)\bigcap C = (A \bigcap C)-B$

4ii) $(A \bigcup B) - (A \bigcap B)=(A-B) \bigcup (B-A)$

4iii) $A-(B-C)=(A-B)\bigcup (A \bigcap B \bigcap C)$

4iv) $(A-B) \times C = (A \times C) - (B \times C)$

V) Let I be a nonempty set and for each $i \in I$ let $X_{i}$ be a set. Prove that

5a) for any set B, we have : $B \bigcap \bigcup_{i \in I} X_{i} = \bigcup_{i \in I}(B \bigcap X_{i})$

5b) if each $X_{i}$ is a subset of a given set S, then $(\bigcup_{i \in I}X_{i})^{'}=\bigcap_{i \in I}(X_{i})^{'}$

VI) Prove that if $f: X \rightarrow Y$ , $g: Y \rightarrow Z$ , and $Z \rightarrow W$ are functions, then : $h \circ (g \circ f) = (h \circ g) \circ f$

VII) Let $f: X \rightarrow Y$ be a function, let A and B be subsets of X, and let C and D be subsets of Y. Prove that:

7i) $f(A \bigcup B) = f(A) \bigcup f(B)$ ; in words, image of union of two sets is the union of two images;

7ii) $f(A \bigcap B) \subset f(A) \bigcap f(B)$ ; in words, image of intersection of two sets is a subset of the intersection of the two images;

7iii) $f^{-1}(C \bigcup D) = f^{-1}(C) \bigcup f^{-1}(D)$ ; in words, the inverse image of the union of two sets is the union of the images of the two sets.

7iv) $f^{-1}(C \bigcap D)=f^{-1}(C) \bigcap f^{-1}(D)$ ; in words, the inverse image of intersection of two sets is intersection of the two inverse images.

7v) $f^{-1}(f(A)) \supset A$ ; in words, the inverse of the image of a set contains the set itself.

7vi) $f(f^{-1}(C)) \subset C$ ; 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 $f: X \rightarrow Y$ 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 $f^{-1}(f(A))=A$ for all $A \subset X$ ; 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 $f(f^{-1}(C))=C$ for all $C \subset Y$ ; 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

Problem:

Let $m_{n}$ be a smallest value of the function $f_{n}(x)=\sum_{k=0}^{2n}x^{k}$ . Prove that $m_{n} \rightarrow \frac{1}{2}$ when $n \rightarrow \infty$ .

Proof:

For $n>1$ ,

$f_{n}(x)=1+x+x^{2}+\ldots=1+x(1+x+x^{2}+x^{4}+\ldots)+x^{2}(1+x^{2}+x^{4}+\ldots)=1+x(1+x)\sum_{k=0}^{n-1}x^{2k}$ .

From this, we see that $f_{n}(x)\geq 1$ for $x \leq -1$ and $x\geq 0$ . Consequently, $f_{n}$ attains its maximum value in the interval $(-1,0)$ . On this interval