mhf-3202-homework-19fall

MHF-3202 Homework

 Homework 7, due Friday 15 November. Section 12.6, problem 10. Suppose $$A,B$$ are sets, $$X,Y\subset B$$ and $$f:A\to B.$$ Show $$f^{-1}(X\cap Y) = f^{-1}(X)\cap f^{-1}(Y).$$ If you prefer, prove the general statement: If $$\mathcal F\subset P(B)$$ is a non-empty set, then $$f^{-1}(\bigcap_{Y\in \mathcal F}Y) = \bigcap_{Y\in \mathcal F} f^{-1}(Y).$$ [Note: It is not assumed that $$f$$ is invertible.] (11 Nov) Latex Code: Suppose $A,B$ are sets, $X,Y\subset B$ and $f:A\to B.$ Show $f^{-1}(X\cap Y) = f^{-1}(X)\cap f^{-1}(Y).$ If you prefer, prove the general statement: If $\mathcal F\subset P(B)$ is a non-empty set, then $f^{-1}(\bigcap_{Y\in \mathcal F}Y) = \bigcap_{Y\in \mathcal F} f^{-1}(Y).$ [Note: It is not assumed that $f$ is invertible.] Homework 6, due Friday 8 November. [A variation of Problem 14, Section 12.2.] Fix a nonempty set $$S$$ and define $$f:\mathcal P(S)\to \mathcal P(S)$$ by $$f(X)=S\setminus X.$$ Show, $$f(f(X))=X$$ for each $$X\in \mathcal P(S).$$ Show $$f$$ is bijective. [A variation of Problem 14, Section 12.4.] Fix a nonempty set $S$ and define $f:\mathcal P(S)\to \mathcal P(S)$ by $f(X)=S\setminus X.$ Show $f(f(X)) = X$ for all $X\in \mathcal P(S).$ Show $f$ is bijective. Homework 5, due Monday 4 November. Show, if $$p\in \mathbb N$$ is prime and $$a$$ is an integer such that $$01,$$, then $$b\mid a$$ or $$b$$ is not prime. Compare with Problem 27, chapter 4. (30 Sep) Suppose $a,b\in\mathbb N.$ Use the contrapositive to show: If $\operatorname{gcd}(a,b)>1,$ then $b\mid a$ or $b$ is not prime. Homework 2, due Wednesday 2 October. Show, if $$n\in\mathbb N$$ and $$n\ge 2$$, then the natural numbers $$n!+2, n!+3, \dots, n!+n$$ are all composite. In the text, the author notes: Thus there are arbitrarily large “gaps” between prime numbers. Express this statement in a precise mathematical form. (27 Sep) Show, if $n\in\mathbb N$ and $n\ge 2$, then the natural numbers $n!+2, n!+3, \dots, n!+n$ are all composite. In the text, the author notes: {\it Thus there are arbitrarily large gaps between prime numbers.} Express this statement in a precise mathematical form. Homework 1, due Friday 27 September. Chapter 4 Problem 6. (23 Sep) In case you are using latex, below is source code for the problem. Suppose $a,b,c\in \mathbb Z$. If $a\mid b$ and $a\mid c,$ then $a\mid (b+c).$ Homework grading abbreviations QU. Quantifier unclear. QE. Quantifier error. QI. The symbol needs an introduction with a quantifier. IT. If needs a then, or vice-versa. IDF. I don’t follow. Circular. Assumes what is to be proved. ETR. Easy to read. Thanks! MC. Missing conjunction(s). SF. Sentence fragment or not a sentence. DNP. Parse error (doesn’t parse). DNF. Does not follow. NN. Narration needed. NRD. Not relevant, delete. ꟼ start new paragraph.