• Syllabus

Topic 4 - Sets, relations and groups

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Sub sections and their related questions

4.1

• 16M.1.hl.TZ0.1a: Explain why only one of the following statements is true (i)     \(17 \subset P\); (ii)    ...
• 17M.1.hl.TZ0.10b: Determine whether or not \(\{ H,{\text{ }} * \} \)  is a subgroup of \(\{ G,{\text{ }} * \} \).
• 18M.1.hl.TZ0.7a: An investigator wishes to test the hypotheses H0 : μ = 65, H1 : μ > 65. He decides on the...
• 18M.1.hl.TZ0.7b.i: She decides to use the same acceptance criteria as the previous investigator. Find the...
• 18M.1.hl.TZ0.7b.ii: Find the critical value for \({\bar x}\) if she wants the probabilities of a Type I error and a...

4.2

• 11M.1.hl.TZ0.3b: For \(a,b \in \mathbb{Z}\) the relation \(aRb\) is defined if and only if \(\frac{a}{b} = {2^k}\)...
• 10M.1.hl.TZ0.2a: Show that \(R\) is an equivalence relation.
• 10M.1.hl.TZ0.2b: The relationship between \(a\) , \(b\) , \(c\) and \(d\) is changed to \(ad - bc = n\) . State,...
• 09M.1.hl.TZ0.1a: Show that \(R\) is an equivalence relation.
• 09M.1.hl.TZ0.1b: State the equivalence classes of \(R\) .
• 13M.2.hl.TZ0.5a: Show that \(R\) is an equivalence relation.
• 13M.2.hl.TZ0.5b: Show, by giving a counter-example, that the statement \({r_1}R{r_2} \Rightarrow r_1^2Rr_2^2\) is...
• 13M.2.hl.TZ0.5c: Determine (i)     the equivalence class \(E\) containing \(1 + \sqrt 2 \) ; (ii)     the...
• 13M.2.hl.TZ0.5d: Show that (i)     \({(1 + \sqrt 2 )^3} \in F\) ; (ii)     \({(1 + \sqrt 2 )^6} \in E\) .
• 08M.2.hl.TZ0.4A.a: The relation \({R_1}\) is defined for \(a,b \in {\mathbb{Z}^ + }\) by \(a{R_1}b\) if and only if...
• 08M.2.hl.TZ0.4B.b: The relation \({R_2}\) is defined for \(a,b \in {\mathbb{Z}^ + }\) by \(a{R_2}b\) if and only if...
• 07M.2.hl.TZ0.5a: The relation \(R\) is defined for \(x,y \in {\mathbb{Z}^ + }\) such that \(xRy\) if and only if...
• SPNone.2.hl.TZ0.1a: Show that \(R\) is an equivalence relation.
• SPNone.2.hl.TZ0.1b: Determine the equivalence class containing \(({x_1},{y_1})\) and interpret it geometrically.
• 14M.1.hl.TZ0.8: The group \(\{ G,{\text{ }} * \} \) has a subgroup \(\{ H,{\text{ }} * \} \). The relation \(R\)...
• 15M.1.hl.TZ0.15a: For \({\rho _1}\) and \({\rho _2}\) determine whether or not each is reflexive, symmetric and...
• 15M.1.hl.TZ0.15b: For each of \({\rho _1}\) and \({\rho _2}\) which is an equivalence relation, describe the...
• 15M.2.hl.TZ0.7b: A relation \(R\) is defined on \(S\) such that \(A\) is related to \(B\) if and only if there...
• 16M.1.hl.TZ0.9b: The relation \(R\) is defined on \({\mathbb{Z}^ + }\) by \(nRm\) if and only if...
• 17M.2.hl.TZ0.2a: The group \(\{ G,{\text{ }} * \} \) has a subgroup \(\{ H,{\text{ }} * \} \). The relation \(R\)...
• 17M.2.hl.TZ0.2b.i: Show that 3\(R\)10.
• 17M.2.hl.TZ0.2b.ii: Determine the three equivalence classes.
• 18M.2.hl.TZ0.6e: Determine the equivalence classes into which R partitions S, giving the first four elements of...
• 18M.2.hl.TZ0.6f: Find two elements in the same equivalence class which are inverses of each other with respect to...

4.3

• 12M.2.hl.TZ0.4A.a: Show that \(f\) is a bijection if \(\boldsymbol{A}\) is non-singular.
• 12M.2.hl.TZ0.4A.b: Suppose now that \(\boldsymbol{A}\) is singular.   (i)     Write down the relationship between...
• 14M.1.hl.TZ0.13: The function...
• 15M.1.hl.TZ0.11: Prove that the function \(f:\mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} \times \mathbb{Z}\)...
• 16M.1.hl.TZ0.1b: (i)     State the value of \(f(1)\), giving a reason for your answer. (ii)     Find...
• 16M.1.hl.TZ0.1c: Determine whether or not \(f\) is (i)     injective; (ii)    surjective.
• 18M.1.hl.TZ0.9a: Given that A is the interval \(\left\{ {x\,{\text{:}}\,0 \leqslant x \leqslant 3} \right\}\) and...
• 18M.1.hl.TZ0.9b.i: Show that the function \(f\) is a bijection.
• 18M.1.hl.TZ0.9b.ii: Hence find the inverse function \({f^{ - 1}}\).

4.4

• 08M.1.hl.TZ0.2a: Draw the Cayley table for \(\left\{ {G,\left.  *  \right\}} \right.\) .
• SPNone.2.hl.TZ0.4a: Copy and complete the following Cayley table.
• 15M.1.hl.TZ0.3a: Construct the Cayley table for \(\{ S,{\text{ }}{ + _6}\} \).

4.5

• 10M.2.hl.TZ0.1a: (i)     Show that \( * \) is associative. (ii)      Find the identity element. (iii)     Find...
• 18M.2.hl.TZ0.6d: Prove that the operation \( \circ \) is not distributive over \( * \).

4.6

• 10M.2.hl.TZ0.1a: (i)     Show that \( * \) is associative. (ii)      Find the identity element. (iii)     Find...
• 16M.2.hl.TZ0.6b: State the identity in \(J\).
• 16M.2.hl.TZ0.6c: Show that (i)     \(1 - \sqrt 2 \) has an inverse in \(J\); (ii)     \(2 + 4\sqrt 2 \) has no...
• 16M.2.hl.TZ0.6d: Show that the subset, \(G\), of elements of \(J\) which have inverses, forms a group of infinite...
• 16M.2.hl.TZ0.6e: (i)     Find the inverse of \(a + b\sqrt 2 \). (ii)     Hence show that \({a^2} - 2{b^2}\)...
• 17M.2.hl.TZ0.8a.i: Show that there are no elements \(a,{\text{ }}b \in {S_n}\) such that \(a{ \times _n}b = 0\).
• 17M.2.hl.TZ0.8a.ii: Show that, for...
• 17M.2.hl.TZ0.8c.ii: Show that the inverse of the element 2 is \(\frac{1}{2}(n + 1)\).
• 17M.2.hl.TZ0.8c.iii: Explain why the inverse of the element 3 is \(\frac{1}{3}(n + 1)\) for some values of \(n\) but...
• 17M.2.hl.TZ0.8c.iv: Determine the inverse of the element 3 in \({G_{11}}\).
• 17M.2.hl.TZ0.8c.v: Determine the inverse of the element 3 in \({G_{31}}\).
• 18M.2.hl.TZ0.6a: Find the identity element of S with respect to \( * \).
• 18M.2.hl.TZ0.6b: Show that every element of S has an inverse with respect to \( * \).
• 18M.2.hl.TZ0.6c: State which elements of S are self-inverse with respect to \( * \).

4.7

• 11M.2.hl.TZ0.6a: (i)     Draw the Cayley table for the set \(S = \left\{ {0,1,2,3,4,\left. 5 \right\}} \right.\)...
• 10M.2.hl.TZ0.1b: The domain of \( * \) is now reduced to \(S = \left\{ {0,2,3,4,5,\left. 6 \right\}} \right.\) and...
• 09M.2.hl.TZ0.2A.a: (i)     Show that \({\mathbb{Z}_4}\) (the set of integers modulo 4) together with the operation...
• 09M.2.hl.TZ0.2B.b: the group is cyclic.
• 13M.1.hl.TZ0.2a: Show that \({(ba)^2} = e\) .
• 13M.1.hl.TZ0.2b: Express \({(bab)^{ - 1}}\) in its simplest form.
• 13M.1.hl.TZ0.2c: Given that \(a \ne e\) ,   (i)     show that \(b \ne e\) ;   (ii)     show that \(G\) is not...
• 13M.2.hl.TZ0.5e: Determine whether the set \(E\) forms a group under   (i)     the operation of addition;   (ii)...
• 08M.1.hl.TZ0.2c: Solve the equation \(x * 6 * x = 3\) where \(x \in G\) .
• 07M.1.hl.TZ0.3a: Show that the set \(S\) of numbers of the form \({2^m} \times {3^n}\) , where...
• 07M.2.hl.TZ0.5b: Let \(S\) denote the set...
• 12M.1.hl.TZ0.1a: The set \({{\rm{S}}_1} = \left\{ {2,4,6,8} \right\}\) and \({ \times _{10}}\) denotes...
• SPNone.1.hl.TZ0.14b: Show that \(S\) forms an Abelian group under matrix multiplication.
• SPNone.2.hl.TZ0.4b: Show that \(\left\{ {S,{ \times _9}} \right\}\) is not a group.
• SPNone.2.hl.TZ0.4c: Prove that a group \(\left\{ {G,{ \times _9}} \right\}\) can be formed by removing two elements...
• SPNone.2.hl.TZ0.4e: Solve the equation \(4{ \times _9}x{ \times _9}x = 1\) .
• 14M.2.hl.TZ0.2a: The set \(S\) contains the eighth roots of unity given by...
• 15M.1.hl.TZ0.3b: Show that \(\{ S,{\text{ }}{ + _6}\} \) forms an Abelian group.
• 15M.2.hl.TZ0.9c: Show that if \(G\) is Abelian, then \(H\) must also be Abelian.
• 16M.2.hl.TZ0.6a: Show that \(J\) is closed.
• 17M.1.hl.TZ0.10a: Show that \(\{ G,{\text{ }} * \} \) is a group.
• 17M.2.hl.TZ0.8b: Show that \({G_n} = \{ {S_n},{\text{ }}{ \times _n}\} \) is a group. You may assume that...
• 18M.1.hl.TZ0.4a: Copy and complete the following Cayley table for the transformations of T1, T2, T3, T4, under the...
• 18M.1.hl.TZ0.4b.i: Show that T1, T2, T3, T4 under the operation of composition of transformations form a group....
• 18M.1.hl.TZ0.4b.ii: Show that this group is cyclic.

4.8

None

4.9

• 11M.2.hl.TZ0.6a: (i)     Draw the Cayley table for the set \(S = \left\{ {0,1,2,3,4,\left. 5 \right\}} \right.\)...
• 11M.2.hl.TZ0.6b: Prove that a cyclic group with exactly one generator cannot have more than two elements.
• 10M.2.hl.TZ0.1b: The domain of \( * \) is now reduced to \(S = \left\{ {0,2,3,4,5,\left. 6 \right\}} \right.\) and...
• 09M.2.hl.TZ0.2b: the group is cyclic.
• 08M.1.hl.TZ0.2b: (i)     Determine the order of each element of \(\left\{ {G,\left.  *  \right\}} \right.\)...
• 12M.1.hl.TZ0.1a: The set \({{\rm{S}}_1} = \left\{ {2,4,6,8} \right\}\) and \({ \times _{10}}\) denotes...
• 12M.2.hl.TZ0.4B.a: Show that \(\left\{ {S,{ + _m}} \right\}\) is cyclic for all m .
• 12M.2.hl.TZ0.4B.b: Given that \(m\) is prime,   (i)     explain why all elements except the identity are generators...
• SPNone.1.hl.TZ0.14c: Giving a reason, state whether or not this group is cyclic.
• SPNone.2.hl.TZ0.4d: (i)     Find the order of all the elements of \(G\) . (ii)     Write down all the proper...
• 14M.2.hl.TZ0.2a: The set \(S\) contains the eighth roots of unity given by...
• 15M.1.hl.TZ0.3c: State the order of each element.
• 15M.1.hl.TZ0.3d: Explain whether or not the group is cyclic.
• 17M.2.hl.TZ0.8c.i: Show that the order of the element \((n - 1)\) is 2.

4.10

• 16M.2.hl.TZ0.8a: (i)     Show that the order of \({S_n}\) is \(n!\); (ii)     List the 6 elements of \({S_3}\) in...
• 16M.2.hl.TZ0.8b: (i)     Write down the matrices M\(_1\), M\(_2\) representing the permutations...
• 16M.2.hl.TZ0.8c: (i)     Use mathematical induction to prove...
• 17M.1.hl.TZ0.6a: Determine the order of \(P\), justifying your answer.
• 17M.1.hl.TZ0.6b: Find \({P^2}\).
• 17M.1.hl.TZ0.6c: The permutation group \(G\) is generated by \(P\). Determine the element of \(G\) that is of...

4.11

• 11M.2.hl.TZ0.6a: (i)     Draw the Cayley table for the set \(S = \left\{ {0,1,2,3,4,\left. 5 \right\}} \right.\)...
• 10M.2.hl.TZ0.1b: The domain of \( * \) is now reduced to \(S = \left\{ {0,2,3,4,5,\left. 6 \right\}} \right.\) and...
• 08M.1.hl.TZ0.2b: (i)     Determine the order of each element of \(\left\{ {G,\left.  *  \right\}} \right.\)...
• 08M.2.hl.TZ0.4B: Consider the group \(\left\{ {G, * } \right\}\) and let \(H\) be a subset of \(G\) defined...
• 12M.2.hl.TZ0.4B.c: Suppose now that \(m = ab\) where \(a\) , \(b\) are unequal prime numbers. Show that...
• SPNone.2.hl.TZ0.4d: (i)     Find the order of all the elements of \(G\) . (ii)     Write down all the proper...
• 14M.1.hl.TZ0.16: \(\{ G,{\text{ }} * \} \) is a group of order \(N\) and \(\{ H,{\text{ }} * \} \) is a proper...
• 15M.2.hl.TZ0.9d: Show that if \(S\) is a subgroup of \(G\), then \(f(S)\) is a subgroup of \(H\).

4.12

• 11M.2.hl.TZ0.6c: \(H\) is a group and the function \(\Phi :H \to H\) is defined by \(\Phi (a) = {a^{ - 1}}\) ,...
• 09M.2.hl.TZ0.2A.b: Using Cayley tables or otherwise, show that \(G\) and...
• 07M.1.hl.TZ0.3b: Show that \(\left\{ {S, \times } \right\}\) is isomorphic to the group of complex numbers...
• 12M.1.hl.TZ0.1b: Now consider the group \(\left\{ {{{\rm{S}}_1},{ \times _{20}}} \right\}\) where...
• SPNone.1.hl.TZ0.6: The group \(\left\{ {G, + } \right\}\) is defined by the operation of addition on the set...
• 14M.2.hl.TZ0.2a: The set \(S\) contains the eighth roots of unity given by...
• 15M.2.hl.TZ0.9a: Show that if \(e\) is the identity in \(G\), then \(f(e)\) is the identity in \(H\).
• 15M.2.hl.TZ0.9b: Show that if \(x\) is an element of \(G\), then \(f({x^{ - 1}}) = {\left( {f(x)} \right)^{ - 1}}\).
• 15M.2.hl.TZ0.9c: Show that if \(G\) is Abelian, then \(H\) must also be Abelian.
• 15M.2.hl.TZ0.9d: Show that if \(S\) is a subgroup of \(G\), then \(f(S)\) is a subgroup of \(H\).