7.1 Workshop 1 (Exploration)

Below, you are given two pairs of objects. For each pair, identify a context in which the objects in that pair are equivalent and a context in which they are not equivalent.

1. (a)
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   and 14:25

2. (b)
   ​

   and

Consider the following mathematical expressions:

1. 1.
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   For each $k\in\{1,2,\ldots,9\}$, define a function $p_{k}(x):X_{k}\to\mathbb{R}$ whose domain $X_{k}$ is the largest possible subset of $\mathbb{R}$ (i.e. the implied domain) and whose rule is given by expression $p_{k}$. Use graph plotting software as needed to sketch the graph (for $p_{6}$ explain how this varies with $n$), and find the intersection point(s) of the graph with the $x$-axis.

2. 2.
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   For each expression $p_{k}$, solve the equation $p_{k}=0$ and find the set of solutions (roots) $S_{k}=\{x\in X_{k}:p_{k}(x)=0\}$. Justify your answer fully algebraically.

3. 3.
   ​

   Hence, group the expressions into sets which:

   1. (a)
      ​

      Define the same function.

   2. (b)
      ​

      Have the same set of solutions.

   3. (c)
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      Have the same solutions if we take multiplicity of roots into account, i.e. if we treat any repeated roots as a separate solution each time that root appears.

Consider the following congruence:

1. 1.
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   Find all other integers which are congruent to $1\mod 18$.

2. 2.
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   When calculating modulo $18$, we generally only use the numbers $0,1,2,\ldots,17$. Why is this sufficient to represent any integer $\mod 18$? More generally, when we calculate modulo $m$, we only use the numbers $0,1,\ldots,m-1$. Why?

3. 3.
   ​

   Find all values $m$ such that 19 is congruent to $1\mod m$. What do these values have in common?

Complete the addition and multiplication tables modulo $5,6$ and $7$. Remember to only use the numbers $0,1,2,\ldots,m-1$ when working modulo $m$.

(Warm Up Task)

1. 1.
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   In light of preparatory tasks 7.1 and 7.2, discuss in your group what it means for two mathematical objects to be equal, equivalent or the same. In general, when are two mathematical expressions equivalent? When are two functions equivalent? When are two equations equivalent?

2. 2.
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   Share the conjectures you made in preparatory Task 7.3 with the others in your group and compare them.

1. 1.
   ​

   In your groups, compare your addition and multiplication tables modulo 5, 6 and 7 from preparatory Task 7.4. What patterns do you notice?

2. 2.
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   Use your multiplication tables to find a counterexample to the following statement.

   “If $ab=0\mod 6$ then $a=0\mod 6$ or $b=0\mod 6$.”

   Can you find a counterexample when working modulo $5$ or $7$?

3. 3.
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   Consider the statement:

   “If $ab=0\mod m$ then $a=0\mod m$ or $b=0\mod m$.”

   For which values of $m$ do we have a counterexample? For which values of $m$ do we NOT have a counterexample? Test your conjectures with some specific examples.

4. 4.
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   How might you define “division” when working modulo $m$?

5. 5.
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   EXTENSION: If $a=b\mod m$, is it true that $a^{n}=b^{n}\mod m$ for all $n\in\mathbb{N}$? When is the converse true? Try and prove your claims.

Use congruences to investigate the following.

1. 1.
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   Let $n\in\mathbb{N}$ and $p\in\mathbb{Z}$.

   1. (a)
      ​

      Find all values $p$ such that

      $$p^{2}=2\mod 3.$$
   2. (b)
      ​

      Hence, determine whether the set $\{3n+8:n\in\mathbb{N}\}$ contains any square numbers.

2. 2.
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   Let $n\in\mathbb{N}$ be expressed by the digits $a_{r}a_{r-1}\ldots a_{0}$, where $a_{0},\ldots,a_{r}\in\{0,1,\ldots,9\}$. Is it true that $n$ is divisible by 3 if and only if the sum of the digits is divisible by 3? Is this divisibility rule true if we replace 3 with any other natural number?

3. 3.
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   EXTENSION: Find the last digit of $7^{7^{1000}}$. Can you generalise your method to find the last digit of $a^{a^{1000}}$ for any integer $a$?

   Hint: Working $\mod 10$, find the first 5 powers of $7$.

The Minister of Transport is planning a public transport system between 10 cities and wants to produce a range of plans for possible selection which obey the following two rules:

1. 1.
   ​

   Each city must have its own public transport system that you can use to get around that city (i.e. each city must be connected to itself).

2. 2.
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   For each pair of cities, the sets of other cities they are connected to via public transport are either identical or disjoint (i.e. they have either no connected cities in common or exactly the same connected cities).

Consider the following grid where dots can be filled in if there is a public transport route from city 1 to city 2. Draw out some possibilities that meet the Minister’s conditions on the following grid:

Compare the different plans you came up with in your group. Can you find a more mathematical way to express the Minister’s rules?

EXTENSION: Using your plans, can you always get between all cities or do you have distinct groups of cities that you cannot move between via public transport? Try proving your claims.

1. 1.
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   Look back at 7.8. What properties does this public transport relation have? Which properties link to which of the Minister’s rules?

2. 2.
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   In your groups, consider the following list of symbols (written in context).

   	$\displaystyle 5$	$\displaystyle<10$	$\displaystyle 5=2$	$\displaystyle\mod 3$	$\displaystyle 6$	$\displaystyle\neq 4$	$\displaystyle 111$	$\displaystyle\in\mathbb{Z}$	$\displaystyle 12$	$\displaystyle=\frac{48}{4}$
   	$\displaystyle\pi$	$\displaystyle\approx 3.14$	$\displaystyle X$	$\displaystyle\subseteq Y$	$\displaystyle\mathbb{Z}$	$\displaystyle\nsubseteq\mathbb{N}$	$\displaystyle 7$	$\displaystyle|91$	$\displaystyle 0$	$\displaystyle\geq-1$

   Which of these symbols represents a relation? What properties do the relations have?

3. 3.
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   EXTENSION: Is it possible to deduce any one of the three properties we’ve been introduced to from the other two?

Consider the set $X=\{1,2,3,4\}$ with the relation $R$. Suppose we know $1R2$ and $2R3$.

1. 1.
   ​

   If $R$ is an equivalence relation, there are only two possibilities for $R$. Find them. Why are there only two possibilities?

2. 2.
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   Suppose $R$ is only symmetric and NOT reflexive or transitive on any $a,b,c\in X$. What more information would you need to know for there to only be one possible relation $R$?

   Hint: Drawing the possible relations will help!

3. 3.
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   Describe the possible relations if $R$ is neither symmetric, reflexive or transitive.