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 nonempty 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 nonempty 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 \( 0<a<p,\) then there is an integer \( b \) such that \( ab\equiv 1 ({\mbox{mod } p}).\) [Suggestion: Observe that \( \gcd(a,p)=1\) and apply Proposition 7.1 on Page 152.] Now show, in \( \mathbb Z_p\), if \( [a]\ne [0],\) then there is a \( [b] \) such that \( [a]\cdot [b]=[1].\) (Thus any nonzero element of \( \mathbb Z_p \) has a multiplicative inverse.) (30 Oct)
Show, if $p\in\mathbb N$ is prime and $a$ is an integer such that $0<a<p$, then there is an integer $b$ such that $ab\equiv 1 ({\mbox{mod } p}).$ [Suggestion: Observe that $\gcd(a,p)=1$ and apply Proposition 7.1 on Page 152.]
Now show, in $\mathbb Z_p$, if $[a]\ne [0]$, then there is a $[b]$ such that $[a]\cdot [b]=[1].$ (Thus any nonzero element of $\mathbb Z_p$ has a multiplicative inverse.)
Homework 4, due Friday 11 October. Chapter 6, Problem 24. (7 Oct)
Homework 3, due Monday 7 October. Suppose \( a,b\in \mathbb N.\) Use the contrapositive to prove: If \(\mbox{gcd}(a,b)>1,\),
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 viceversa.
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.
