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Topic 8 - Option: Sets, relations and groups

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Topic 8 - Option: Sets, relations and groups

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The aims of this option are to provide the opportunity to study some important mathematical concepts, and introduce the principles of proof through abstract algebra.

Directly related questions

18M.3srg.hl.TZ0.1a: Show that {T, ×10} is a group. (You may assume associativity.)
18M.3srg.hl.TZ0.5b.ii: Show that \(f\) is surjective.
18M.3srg.hl.TZ0.5b.i: Show that \(f\) is injective.
18M.3srg.hl.TZ0.5a: Prove that \(f \circ f\) is the identity function.
18M.3srg.hl.TZ0.4c: Let \(f{\text{:}}\,{S_4} \to {S_4}\) be defined by \(f\left( p \right) = p \circ p\)...
18M.3srg.hl.TZ0.4b: Find the proper subgroup H of order 6 containing \({p_1}\), \({p_2}\) and their compositions....
18M.3srg.hl.TZ0.4a: Determine the order of S4.
18M.3srg.hl.TZ0.3b: Show, by means of an example, that \(R\) is not transitive.
18M.3srg.hl.TZ0.3a.ii: Show that \(R\) is symmetric.
18M.3srg.hl.TZ0.3a.i: Show that \(R\) is reflexive.
18M.3srg.hl.TZ0.2b: Let S be a set containing \(n\) elements where \(n \in \mathbb{N}\). Show that S has \({2^n}\)...
18M.3srg.hl.TZ0.2a.ii: Verify that A \ C ≠ C \ A.
18M.3srg.hl.TZ0.2a.i: Find \(\left( {A \cup B} \right) \cap \left( {A \cup C} \right)\).
18M.3srg.hl.TZ0.1d: The binary operation multiplication modulo 10, denoted by ×10 , is defined on the set V = {1, 3...
18M.3srg.hl.TZ0.1c.ii: Hence show that {T, ×10} is cyclic and write down all its generators.
18M.3srg.hl.TZ0.1c.i: Find the order of each element of {T, ×10}.
18M.3srg.hl.TZ0.1b: By making reference to the Cayley table, explain why T is Abelian.
16M.3srg.hl.TZ0.5: The group \(\{ G,{\text{ }} * \} \) is defined on the set \(G\) with binary operation \( * \)....
16M.3srg.hl.TZ0.3: The group \(\{ G,{\text{ }} * \} \) is Abelian and the bijection \(f:{\text{ }}G \to G\) is...
16M.3srg.hl.TZ0.1e: Solve the equation \(2 * x * 4 * x * 4 = 2\).
16M.3srg.hl.TZ0.1d: (i) Find the two proper subgroups of \(\{ S,{\text{ }} * \} \). (ii) Find the coset of...
16M.3srg.hl.TZ0.1b: Show that \(\{ S,{\text{ }} * \} \) is an Abelian group.
17M.3srg.hl.TZ0.2a.i: Show that \(R\) is an equivalence relation.
16N.3srg.hl.TZ0.4c: Hence prove that there is at least one element of \(A\) that is not related to any other element...
16N.3srg.hl.TZ0.4b: Explain why there exists an element \(a \in A\) that is not related to itself.
16N.3srg.hl.TZ0.4a: Show that \(S\) is (i) not reflexive; (ii) symmetric; (iii) transitive.
16N.3srg.hl.TZ0.3b: (i) State a generator for \(\{ H,{\text{ }} * \} \). (ii) Write down the elements of...
16N.3srg.hl.TZ0.3a: State the possible orders of an element of \(\{ G,{\text{ }} * \} \) and for each order give an...
16N.3srg.hl.TZ0.2b: (i) Sketch the graph of \(y = g(x)\) and hence justify whether or not \(g\) is a...
16N.3srg.hl.TZ0.2a: (i) Sketch the graph of \(y = f(x)\) and hence justify whether or not \(f\) is a...
16N.3srg.hl.TZ0.1f: Find the number of permutations in \(\{ G,{\text{ }} \circ \} \) which will result in \(A\),...
16N.3srg.hl.TZ0.1e: State the order of \(\{ G,{\text{ }} \circ \} \).
16N.3srg.hl.TZ0.1d: Write the permutation \(\beta \circ \alpha \) as a composition of disjoint cycles.
16N.3srg.hl.TZ0.1c: Write the permutation \(\alpha \circ \beta \) as a composition of disjoint cycles.
16N.3srg.hl.TZ0.1b: (i) Write the permutation...
16N.3srg.hl.TZ0.1a: (i) Write the permutation...
16M.3srg.hl.TZ0.1a: Copy and complete the table.
16M.3srg.hl.TZ0.1c: Determine the orders of all the elements of \(\{ S,{\text{ }} * \} \).
16M.3srg.hl.TZ0.2a: Show that \(R\) is an equivalence relation.
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.TZ0.4a: Prove that \(f\) is an injection.
16M.3srg.hl.TZ0.4b: (i) Prove that \(f\) is a surjection. (ii) Hence, or otherwise, write down the inverse...
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 illustrate...
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 \(P \cup (Q\Delta R) \ne (P \cup Q)\Delta (P \cup R)\).
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 the...
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 \( \cap \) is...
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 only if...
15N.3srg.hl.TZ0.2b: The relation \(R\) is defined for \(a,{\text{ }}b \in \mathbb{R}\) so that \(aRb\) if and only if...
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, t} is...
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 = hg...
12N.3srg.hl.TZ0.1a: Decide, giving a proof or a counter-example, whether \(xRy \Leftrightarrow x + y > 7\) is (i)...
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 \(xRy \Leftrightarrow xy > 0\) is (i) ...
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 assume...
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 codomain...
08M.3srg.hl.TZ1.3: (a) Find the six roots of the equation \({z^6} - 1 = 0\) , giving your answers in the form...
08M.3srg.hl.TZ1.4: (a) Show that R is an equivalence relation. (b) Describe, geometrically, the equivalence...
08M.3srg.hl.TZ1.5: Let \(p = {2^k} + 1,{\text{ }}k \in {\mathbb{Z}^ + }\) be a prime number and let G be the group...
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 \(F:P \to Q{\text{ and }}g:A \to B\) . Determine,...
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 k...
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 a...
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 multiplication. (b) ...
09M.3srg.hl.TZ0.2a: (i) Show that \( * \) is commutative. (ii) Find the identity element. (iii) Find...
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 by...
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) Determine...
SPNone.3srg.hl.TZ0.3b: The group \(\{ K,{\text{ }} \circ \} \) is defined on the six permutations of the integers 1, 2,...
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 that...
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 \(aRb \Leftrightarrow ab \geqslant 0\),...
10N.3srg.hl.TZ0.2a: Let \(f:\mathbb{Z} \times \mathbb{R} \to \mathbb{R},{\text{ }}f(m,{\text{ }}x) = {( - 1)^m}x\)....
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 \(H \cap K\)...
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 isomorphic.
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 disjoint sets.
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) and...
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 \(3{s_1} + 5{s_2} \in...
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 group...
14M.3srg.hl.TZ0.1: The binary operation \(\Delta\) is defined on the set \(S =\) {1, 2, 3, 4, 5} by the following...
14M.3srg.hl.TZ0.3b: Consider the function \(f:X \times Y \to \mathbb{R}\) defined by \(f(x,{\text{ }}y) = x + y\) and...
14M.3srg.hl.TZ0.4c: (i) Prove that \(gk{g^{ - 1}} \in K\) for all \(g \in G,{\text{ }}k \in K\). (ii) Deduce...
13N.3srg.hl.TZ0.2a: (i) Prove that \(G\) is cyclic and state two of its generators. (ii) Let \(H\) be the...
13N.3srg.hl.TZ0.2b: State, with a reason, whether or not it is necessary that a group is cyclic given that all its...
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 \({e_H}\)...
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 \((S,{\text{ }} * )\) is...
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 pair...
15M.3srg.hl.TZ0.5a: Show that \((G,{\text{ }} + )\) forms a group where \( + \) denotes addition on \(\mathbb{Q}\)....
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 \(R\) be...
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 \(R\) be...
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 \(R\) be...

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Topic 8 - Option: Sets, relations and groups

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