Date | May 2016 | Marks available | 7 | Reference code | 16M.3srg.hl.TZ0.2 |
Level | HL only | Paper | Paper 3 Sets, relations and groups | Time zone | TZ0 |
Command term | Show that | Question number | 2 | Adapted from | N/A |
Question
The relation R= is defined on Z+ such that aRb if and only if bn−an≡0(mod where n,{\text{ }}p are fixed positive integers greater than 1.
Show that R is an equivalence relation.
Given that n = 2 and p = 7, determine the first four members of each of the four equivalence classes of R.
Markscheme
since {a^n} - {a^n} = 0, A1
it follows that (aRa) and R is reflexive R1
if aRb so that {b^n} - {a^n} \equiv 0(\bmod p) M1
then, {a^n} - {b^n} \equiv 0(\bmod p) so that bRa and R is symmetric A1
if aRb and bRc so that {b^n} - {a^n} \equiv 0(\bmod p) and {c^n} - {b^n} \equiv 0(\bmod p) M1
adding, (it follows that {c^n} - {a^n} \equiv 0(\bmod p)) M1
so that aRc and R is transitive A1
Note: Only accept the correct use of the terms “reflexive, symmetric, transitive”.
[7 marks]
we are now given that aRb if {b^2} - {a^2} \equiv 0(\bmod 7)
attempt to find at least one equivalence class (M1)
the equivalence classes are
\{ 1,{\text{ }}6,{\text{ }}8,{\text{ }}13,{\text{ }} \ldots \} A1
\{ 2,{\text{ }}5,{\text{ }}9,{\text{ }}12,{\text{ }} \ldots \} A1
\{ 3,{\text{ }}4,{\text{ }}10,{\text{ }}11,{\text{ }} \ldots \} A1
\{ 7,{\text{ }}14,{\text{ }}21,{\text{ }}28,{\text{ }} \ldots \} A1
[5 marks]
Examiners report
Most candidates were familiar with the terminology of the required conditions to be satisfied for a relation to be an equivalence relation. The execution of the proofs was variable. It was grating to see such statements as R is symmetric because aRb = bRa or aRa = {a^n} - {a^n} = 0, often without mention of \bmod p, and such responses were not fully rewarded.
This was not well answered. Few candidates displayed a strategy to find the equivalence classes.
Syllabus sections
- 16M.3srg.hl.TZ0.1a: Copy and complete the table.
- 16M.3srg.hl.TZ0.1b: Show that \{ S,{\text{ }} * \} is an Abelian group.
- 16M.3srg.hl.TZ0.1c: Determine the orders of all the elements of \{ S,{\text{ }} * \} .
- 16M.3srg.hl.TZ0.1d: (i) Find the two proper subgroups of \{ S,{\text{ }} * \} . (ii) Find the coset...
- 16M.3srg.hl.TZ0.1e: Solve the equation 2 * x * 4 * x * 4 = 2.
- 16M.3srg.hl.TZ0.2b: Given that n = 2 and p = 7, determine the first four members of each of the four...
- 16M.3srg.hl.TZ0.3: The group \{ G,{\text{ }} * \} is Abelian and the bijection f:{\text{ }}G \to G is...
- 16M.3srg.hl.TZ0.4a: Prove that f is an injection.
- 16M.3srg.hl.TZ0.5: The group \{ G,{\text{ }} * \} is defined on the set G with binary operation \( *...
- 16M.3srg.hl.TZ0.4b: (i) Prove that f is a surjection. (ii) Hence, or otherwise, write down the...
- 17N.3srg.hl.TZ0.5b: Prove that {\text{Ker}}(f) is a subgroup of \{ G,{\text{ }} * \} .
- 17N.3srg.hl.TZ0.5a: Prove that f({e_G}) = {e_H}.
- 17N.3srg.hl.TZ0.4d: Show that each element a \in S has an inverse.
- 17N.3srg.hl.TZ0.4c: Show that 2 is the identity element.
- 17N.3srg.hl.TZ0.4b.i: Show that the operation * on the set S is commutative.
- 17N.3srg.hl.TZ0.4a: Show that x * y \in S for all x,{\text{ }}y \in S.
- 17N.3srg.hl.TZ0.3b: Determine the equivalence class of R containing the element (1,{\text{ }}2) and...
- 17N.3srg.hl.TZ0.3a: Show that R is an equivalence relation.
- 17N.3srg.hl.TZ0.2b.ii: In the context of the distributive law, describe what the result in part (b)(i) illustrates.
- 17N.3srg.hl.TZ0.2b.i: For sets P, Q and R, verify that...
- 17N.3srg.hl.TZ0.2a.ii: Represent the following set on a Venn diagram, A \cap (B \cup C).
- 17N.3srg.hl.TZ0.2a.i: Represent the following set on a Venn diagram, A\Delta B, the symmetric difference of...
- 17N.3srg.hl.TZ0.1c: Find the left cosets of K in \{ G,{\text{ }}{ \times _{18}}\} .
- 17N.3srg.hl.TZ0.1b: Write down the elements in set K.
- 17N.3srg.hl.TZ0.1a.ii: State whether or not \{ G,{\text{ }}{ \times _{18}}\} is cyclic, justifying your answer.
- 17N.3srg.hl.TZ0.1a.i: Find the order of elements 5, 7 and 17 in \{ G,{\text{ }}{ \times _{18}}\} .
- 17M.3srg.hl.TZ0.4d: Show that the groups \{ \mathbb{Z},{\text{ }} * \} and...
- 17M.3srg.hl.TZ0.4c: Find a proper subgroup of \{ \mathbb{Z},{\text{ }} * \} .
- 17M.3srg.hl.TZ0.4b: Show that there is no element of order 2.
- 17M.3srg.hl.TZ0.4a: Show that \{ \mathbb{Z},{\text{ }} * \} is an Abelian group.
- 17M.3srg.hl.TZ0.3b: Hence write down the inverse function {f^{ - 1}}(x,{\text{ }}y).
- 17M.3srg.hl.TZ0.3a: Show that f is a bijection.
- 17M.3srg.hl.TZ0.2b: Determine the number of equivalence classes of S.
- 17M.3srg.hl.TZ0.2a.ii: Determine the equivalence classes of R.
- 17M.3srg.hl.TZ0.1b.ii: Hence by considering A \cap (B \cup C), verify that in this case the operation \cap ...
- 17M.3srg.hl.TZ0.1b.i: Write down all the elements of A \cap B,{\text{ }}A \cap C and B \cup C.
- 17M.3srg.hl.TZ0.1a.ii: Determine the symmetric difference, A\Delta B, of the sets A and B.
- 17M.3srg.hl.TZ0.1a.i: Write down all the elements of A and all the elements of B.
- 15N.3srg.hl.TZ0.5c: Prove that \{ {\text{Ker}}(f),{\text{ }} + \} is a subgroup of...
- 15N.3srg.hl.TZ0.5b: Find the kernel of f.
- 15N.3srg.hl.TZ0.5a: Prove that the function f is a homomorphism from the group...
- 15N.3srg.hl.TZ0.4c: Find the order of each element in T.
- 15N.3srg.hl.TZ0.4b: Prove that \{ T,{\text{ }} * \} forms an Abelian group.
- 15N.3srg.hl.TZ0.4a: Copy and complete the following Cayley table for \{ T,{\text{ }} * \} .
- 15N.3srg.hl.TZ0.3d: (i) Find the maximum possible order of an element in \{ H,{\text{ }} \circ \} . (ii)...
- 15N.3srg.hl.TZ0.3c: Find (i) p \circ p; (ii) the inverse of p \circ p.
- 15N.3srg.hl.TZ0.3b: State the identity element in \{ G,{\text{ }} \circ \} .
- 15N.3srg.hl.TZ0.3a: Find the order of \{ G,{\text{ }} \circ \} .
- 15N.3srg.hl.TZ0.2c: The relation R is defined for a,{\text{ }}b \in \mathbb{R} so that aRb if and...
- 15N.3srg.hl.TZ0.2b: The relation R is defined for a,{\text{ }}b \in \mathbb{R} so that aRb if and...
- 15N.3srg.hl.TZ0.1: Given the sets A and B, use the properties of sets to prove that...
- 12M.3srg.hl.TZ0.1a: Associativity and commutativity are two of the five conditions for a set S with the binary...
- 12M.3srg.hl.TZ0.1b: The Cayley table for the binary operation \odot defined on the set T = {p, q, r, s,...
- 12M.3srg.hl.TZ0.2a: Given that R = (P \cap Q')' , list the elements of R .
- 12M.3srg.hl.TZ0.2b: For a set S , let {S^ * } denote the set of all subsets of S , (i) find \({P^ *...
- 12M.3srg.hl.TZ0.3a: Show that R is an equivalence relation.
- 12M.3srg.hl.TZ0.3b: Find the equivalence class containing 0.
- 12M.3srg.hl.TZ0.3c: Denote the equivalence class containing n by Cn . List the first six elements of {C_1}.
- 12M.3srg.hl.TZ0.3d: Denote the equivalence class containing n by Cn . Prove that {C_n} = {C_{n + 7}} for all...
- 12M.3srg.hl.TZ0.4b: The set S is finite. If the function f:S \to S is injective, show that f is surjective.
- 12M.3srg.hl.TZ0.5a: (i) Show that gh{g^{ - 1}} has order 2 for all g \in G. (ii) Deduce that gh...
- 12N.3srg.hl.TZ0.1a: Decide, giving a proof or a counter-example, whether xRy \Leftrightarrow x + y > 7...
- 12N.3srg.hl.TZ0.1d: Decide, giving a proof or a counter-example, whether...
- 12N.3srg.hl.TZ0.1b: Decide, giving a proof or a counter-example, whether...
- 12N.3srg.hl.TZ0.1c: Decide, giving a proof or a counter-example, whether xRy \Leftrightarrow xy > 0...
- 12N.3srg.hl.TZ0.3e: (i) State the identity element for \{ P(S){\text{, }}\Delta \} . (ii) Write down...
- 12N.3srg.hl.TZ0.1e: One of the relations from parts (a), (b), (c) and (d) is an equivalence relation. For this...
- 12N.3srg.hl.TZ0.3a: Write down all four subsets of A .
- 12N.3srg.hl.TZ0.3b: Construct the Cayley table for P(A) under \Delta .
- 12N.3srg.hl.TZ0.3c: Prove that \left\{ {P(A),{\text{ }}\Delta } \right\} is a group. You are allowed to...
- 12N.3srg.hl.TZ0.3d: Is \{ P(A){\text{, }}\Delta \} isomorphic to \{ {\mathbb{Z}_4},{\text{ }}{ + _4}\} ...
- 12N.3srg.hl.TZ0.3f: Explain why \{ P(S){\text{, }} \cup \} is not a group.
- 12N.3srg.hl.TZ0.3g: Explain why \{ P(S){\text{, }} \cap \} is not a group.
- 12N.3srg.hl.TZ0.4a: Simplify \frac{c}{2} * \frac{{3c}}{4} .
- 12N.3srg.hl.TZ0.4b: State the identity element for G under * .
- 12N.3srg.hl.TZ0.4c: For x \in G find an expression for {x^{ - 1}} (the inverse of x under * ).
- 12N.3srg.hl.TZ0.4d: Show that the binary operation * is commutative on G .
- 12N.3srg.hl.TZ0.4e: Show that the binary operation * is associative on G .
- 12N.3srg.hl.TZ0.4g: Show that G is closed under * .
- 12N.3srg.hl.TZ0.4h: Explain why \{ G, * \} is an Abelian group.
- 08M.3srg.hl.TZ1.1: (a) Determine whether or not * is (i) closed, (ii) commutative, (iii) ...
- 08M.3srg.hl.TZ1.2: (a) Find the range of f . (b) Prove that f is an injection. (c) Taking the...
- 08M.3srg.hl.TZ1.3: (a) Find the six roots of the equation {z^6} - 1 = 0 , giving your answers in the...
- 08M.3srg.hl.TZ1.4: (a) Show that R is an equivalence relation. (b) Describe, geometrically, the...
- 08M.3srg.hl.TZ1.5: Let p = {2^k} + 1,{\text{ }}k \in {\mathbb{Z}^ + } be a prime number and let G be the...
- 08M.3srg.hl.TZ2.1: (a) Draw the Cayley table for the set of integers G = {0, 1, 2, 3, 4, 5} under addition...
- 08M.3srg.hl.TZ2.2a: Below are the graphs of the two functions F:P \to Q{\text{ and }}g:A \to B...
- 08M.3srg.hl.TZ2.2b: Given two functions h:X \to Y{\text{ and }}k:Y \to Z . Show that (i) if both h and...
- 08M.3srg.hl.TZ2.4a: The relation aRb is defined on {1, 2, 3, 4, 5, 6, 7, 8, 9} if and only if ab is the square of...
- 08M.3srg.hl.TZ2.4b: Given the group (G,{\text{ }} * ), a subgroup (H,{\text{ }} * ) and...
- 08M.3srg.hl.TZ2.5: (a) Write down why the table below is a Latin...
- 08M.3srg.hl.TZ2.3: Prove that (A \cap B)\backslash (A \cap C) = A \cap (B\backslash C) where A, B and C are...
- 08N.3srg.hl.TZ0.1: A, B, C and D are subsets of \mathbb{Z}...
- 08N.3srg.hl.TZ0.2: A binary operation is defined on {−1, 0, 1}...
- 08N.3srg.hl.TZ0.5: Three functions mapping \mathbb{Z} \times \mathbb{Z} \to \mathbb{Z} are defined...
- 08N.3srg.hl.TZ0.3: Two functions, F and G , are defined on A = \mathbb{R}\backslash \{ 0,{\text{ }}1\} ...
- 08N.3srg.hl.TZ0.4: Determine, giving reasons, which of the following sets form groups under the operations given...
- 11M.3srg.hl.TZ0.1a: Copy and complete the following operation table.
- 11M.3srg.hl.TZ0.1b: (i) Show that {S , * } is a group. (ii) Find the order of each element of {S ,...
- 11M.3srg.hl.TZ0.1c: The set T is defined by \{ x * x:x \in S\} . Show that {T , * } is a subgroup of {S...
- 11M.3srg.hl.TZ0.2a: A \ B ;
- 11M.3srg.hl.TZ0.3a: Show that R is an equivalence relation.
- 11M.3srg.hl.TZ0.4: The function...
- 11M.3srg.hl.TZ0.5b: Consider the group G , of order 4, which has distinct elements a , b and c and the identity...
- 11M.3srg.hl.TZ0.2b: A\Delta B .
- 11M.3srg.hl.TZ0.3b: Identify the three equivalence classes.
- 11M.3srg.hl.TZ0.5a: Given that p , q and r are elements of a group, prove the left-cancellation rule, i.e....
- 09M.3srg.hl.TZ0.1: (a) Show that {1, −1, i, −i} forms a group of complex numbers G under...
- 09M.3srg.hl.TZ0.2a: (i) Show that * is commutative. (ii) Find the identity element. (iii) ...
- 09M.3srg.hl.TZ0.2b: The binary operation \cdot is defined on \mathbb{R} as follows. For any elements a...
- 09M.3srg.hl.TZ0.3: The relation R is defined on \mathbb{Z} \times \mathbb{Z} such that...
- 09M.3srg.hl.TZ0.4: (a) Show that f:\mathbb{R} \times \mathbb{R} \to \mathbb{R} \times \mathbb{R} defined...
- 09M.3srg.hl.TZ0.5: Prove that set difference is not associative.
- 09N.3srg.hl.TZ0.1: The binary operation * is defined on the set S = {0, 1, 2, 3}...
- 09N.3srg.hl.TZ0.2: The function f:[0,{\text{ }}\infty [ \to [0,{\text{ }}\infty [ is defined by...
- 09N.3srg.hl.TZ0.3: The relations R and S are defined on quadratic polynomials P of the...
- 09N.3srg.hl.TZ0.5: Let {G , * } be a finite group of order n and let H be a non-empty subset of G . (a) ...
- SPNone.3srg.hl.TZ0.1a: The relation R is defined on {\mathbb{Z}^ + } by aRb if and only if ab is even. Show that...
- SPNone.3srg.hl.TZ0.1b: The relation S is defined on {\mathbb{Z}^ + } by aSb if and only if...
- SPNone.3srg.hl.TZ0.2c: * is distributive over \odot ;
- SPNone.3srg.hl.TZ0.2a: \odot is commutative;
- SPNone.3srg.hl.TZ0.2b: * is associative;
- SPNone.3srg.hl.TZ0.2d: \odot has an identity element.
- SPNone.3srg.hl.TZ0.3a: (i) Write down the Cayley table for \{ G,{\text{ }}{ \times _7}\} . (ii) ...
- SPNone.3srg.hl.TZ0.3b: The group \{ K,{\text{ }} \circ \} is defined on the six permutations of the integers 1,...
- SPNone.3srg.hl.TZ0.4: The groups \{ K,{\text{ }} * \} and \{ H,{\text{ }} \odot \} are defined by the...
- SPNone.3srg.hl.TZ0.5: Let \{ G,{\text{ }} * \} be a finite group and let H be a non-empty subset of G . Prove...
- 10M.3srg.hl.TZ0.3: (a) Consider the set A = {1, 3, 5, 7} under the binary operation * , where * ...
- 10M.3srg.hl.TZ0.4: The permutation {p_1} of the set {1, 2, 3, 4} is defined...
- 10M.3srg.hl.TZ0.5: Let G be a finite cyclic group. (a) Prove that G is Abelian. (b) Given that a is a...
- 10M.3srg.hl.TZ0.1: The function f:\mathbb{R} \to \mathbb{R} is defined...
- 10N.3srg.hl.TZ0.1: Let R be a relation on the set \mathbb{Z} such that...
- 10N.3srg.hl.TZ0.2a: Let...
- 10N.3srg.hl.TZ0.2b: P is the set of all polynomials such that...
- 10N.3srg.hl.TZ0.2c: Let h:\mathbb{Z} \to {\mathbb{Z}^ + },...
- 10N.3srg.hl.TZ0.4: Set...
- 10N.3srg.hl.TZ0.5: Let \{ G,{\text{ }} * \} be a finite group that contains an element a (that is not the...
- 13M.3srg.hl.TZ0.1b: is commutative;
- 13M.3srg.hl.TZ0.1c: is associative;
- 13M.3srg.hl.TZ0.2a: Copy and complete the following Cayley table for this binary operation.
- 13M.3srg.hl.TZ0.2c: Show that a new set G can be formed by removing one of the elements of S such that...
- 13M.3srg.hl.TZ0.2d: Determine the order of each element of \{ G,{\text{ }}{ \times _{14}}\} .
- 13M.3srg.hl.TZ0.2e: Find the proper subgroups of \{ G,{\text{ }}{ \times _{14}}\} .
- 13M.3srg.hl.TZ0.4b: Show that the equivalence defining R can be written in the...
- 13M.3srg.hl.TZ0.4c: Hence, or otherwise, determine the equivalence classes.
- 13M.3srg.hl.TZ0.1a: is closed;
- 13M.3srg.hl.TZ0.1d: has an identity element.
- 13M.3srg.hl.TZ0.2b: Give one reason why \{ S,{\text{ }}{ \times _{14}}\} is not a group.
- 13M.3srg.hl.TZ0.3a: (i) Sketch the graph of f. (ii) By referring to your graph, show that f is a bijection.
- 13M.3srg.hl.TZ0.3b: Find {f^{ - 1}}(x).
- 13M.3srg.hl.TZ0.4a: Show that R is an equivalence relation.
- 13M.3srg.hl.TZ0.5: H and K are subgroups of a group G. By considering the four group axioms, prove that...
- 11N.3srg.hl.TZ0.1a: Consider the following Cayley table for the set G = {1, 3, 5, 7, 9, 11, 13, 15} under the...
- 11N.3srg.hl.TZ0.1b: The Cayley table for the set...
- 11N.3srg.hl.TZ0.1c: Show that \{ G,{\text{ }}{ \times _{16}}\} and \{ H,{\text{ }} * \} are not...
- 11N.3srg.hl.TZ0.1d: Show that \{ H,{\text{ }} * \} is not cyclic.
- 11N.3srg.hl.TZ0.2a: Determine, using Venn diagrams, whether the following statements are true. (i) ...
- 11N.3srg.hl.TZ0.2b: Prove, without using a Venn diagram, that A\backslash B and B\backslash A are...
- 11N.3srg.hl.TZ0.4a: Show that R is an equivalence relation.
- 11N.3srg.hl.TZ0.4b: The Cayley table for G is shown below. The subgroup H is given as...
- 11N.3srg.hl.TZ0.5a: Show that if both f and g are injective, then g \circ f is also injective.
- 11N.3srg.hl.TZ0.5b: Show that if both f and g are surjective, then g \circ f is also surjective.
- 11N.3srg.hl.TZ0.5c: Show, using a single counter example, that both of the converses to the results in part (a)...
- 12M.3srg.hl.TZ0.4a: The function g:\mathbb{Z} \to \mathbb{Z} is defined by...
- 12M.3srg.hl.TZ0.4c: Using the set {\mathbb{Z}^ + } as both domain and codomain, give an example of an...
- 14M.3srg.hl.TZ0.2a: (i) Write down the six smallest non-negative elements of S. (ii) Show that...
- 14M.3srg.hl.TZ0.2b: The relation R is defined on S by {s_1}R{s_2} if...
- 14M.3srg.hl.TZ0.3a: (i) Sketch the set X \times Y in the Cartesian plane. (ii) Sketch the set...
- 14M.3srg.hl.TZ0.4b: (i) Prove that the kernel of f,{\text{ }}K = {\text{Ker}}(f), is closed under the...
- 14M.3srg.hl.TZ0.1: The binary operation \Delta is defined on the set S = {1, 2, 3, 4, 5} by the...
- 14M.3srg.hl.TZ0.3b: Consider the function f:X \times Y \to \mathbb{R} defined by f(x,{\text{ }}y) = x + y...
- 14M.3srg.hl.TZ0.4c: (i) Prove that gk{g^{ - 1}} \in K for all g \in G,{\text{ }}k \in K. (ii) ...
- 13N.3srg.hl.TZ0.2a: (i) Prove that G is cyclic and state two of its generators. (ii) Let H be...
- 13N.3srg.hl.TZ0.2b: State, with a reason, whether or not it is necessary that a group is cyclic given that all...
- 13N.3srg.hl.TZ0.1: Consider the following functions ...
- 13N.3srg.hl.TZ0.4: Let (H,{\text{ }} * {\text{)}} be a subgroup of the group \((G,{\text{ }} *...
- 13N.3srg.hl.TZ0.5: (a) Given a set U, and two of its subsets A and B, prove...
- 14M.3srg.hl.TZ0.4a: Prove that f({e_G}) = {e_H}, where {e_G} is the identity element in G and...
- 15M.3srg.hl.TZ0.3b: Hence prove that R is reflexive.
- 15M.3srg.hl.TZ0.4a: Prove that: (i) f is an injection, (ii) g is a surjection.
- 15M.3srg.hl.TZ0.2a: Find the element e such that e * y = y, for all y \in S.
- 15M.3srg.hl.TZ0.2b: (i) Find the least solution of x * x = e. (ii) Deduce that (S,{\text{ }} * )...
- 15M.3srg.hl.TZ0.3d: Find the set of all y for which 3Ry.
- 15M.3srg.hl.TZ0.3e: Using your answers for (c) and (d) show that R is not symmetric.
- 15M.3srg.hl.TZ0.1a: Complete the following Cayley table [5 marks]
- 15M.3srg.hl.TZ0.1b: (i) State the inverse of each element. (ii) Determine the order of each element.
- 15M.3srg.hl.TZ0.1c: Write down the subgroups containing (i) r, (ii) u.
- 15M.3srg.hl.TZ0.2c: Determine whether or not e is an identity element.
- 15M.3srg.hl.TZ0.3a: Show that the product of three consecutive integers is divisible by 6.
- 15M.3srg.hl.TZ0.3c: Find the set of all y for which 5Ry.
- 15M.3srg.hl.TZ0.4b: Given that X = {\mathbb{R}^ + } \cup \{ 0\} and Y = \mathbb{R}, choose a suitable...
- 15M.3srg.hl.TZ0.5a: Show that (G,{\text{ }} + ) forms a group where + denotes addition on...
- 15M.3srg.hl.TZ0.5b: Assuming that (H,{\text{ }} + ) forms a group, show that it is a proper subgroup of...
- 15M.3srg.hl.TZ0.5c: The mapping \phi :G \to G is given by \phi (g) = g + g, for g \in G. Prove that...
- 14N.3srg.hl.TZ0.1b: Find the order of each of the elements of the group.
- 14N.3srg.hl.TZ0.1c: Write down the three sets that form subgroups of order 2.
- 14N.3srg.hl.TZ0.5c: Let \{ H,{\rm{ }} * {\rm{\} }} be a subgroup of \{ G,{\rm{ }} * {\rm{\} }}. Let R...
- 14N.3srg.hl.TZ0.1a: Find the values represented by each of the letters in the table.
- 14N.3srg.hl.TZ0.1d: Find the three sets that form subgroups of order 4.
- 14N.3srg.hl.TZ0.2b: Prove that f is not a surjection.
- 14N.3srg.hl.TZ0.2a: Prove that f is an injection.
- 14N.3srg.hl.TZ0.3a: Two members of A are given by p = (1{\text{ }}2{\text{ }}5) and...
- 14N.3srg.hl.TZ0.4c: Given that f(x * y) = p, find f(y).
- 14N.3srg.hl.TZ0.3b: State a permutation belonging to A of order (i) 4; (ii) 6.
- 14N.3srg.hl.TZ0.3c: Let P = {all permutations in A where exactly two integers change position}, and...
- 14N.3srg.hl.TZ0.4a: Prove that for all a \in G,{\text{ }}f({a^{ - 1}}) = {\left( {f(a)} \right)^{ - 1}}.
- 14N.3srg.hl.TZ0.4b: Let \{ H,{\text{ }} \circ \} be the cyclic group of order seven, and let p be a...
- 14N.3srg.hl.TZ0.5a: State Lagrange’s theorem.
- 14N.3srg.hl.TZ0.5d: Let \{ H,{\rm{ }} * {\rm{\} }} be a subgroup of \{ G,{\rm{ }} * {\rm{\} }} .Let R...
- 14N.3srg.hl.TZ0.5b: Verify that the inverse of a * {b^{ - 1}} is equal to b * {a^{ - 1}}.
- 14N.3srg.hl.TZ0.5e: Let \{ H,{\rm{ }} * {\rm{\} }} be a subgroup of \{ G,{\rm{ }} * {\rm{\} }} .Let R...