DP Further Mathematics HL Questionbank

Topic 4 - Sets, relations and groups
Description
The aims of this topic are to provide the opportunity to study some important mathematical concepts, and introduce the principles of proof through abstract algebra.
Directly related questions
- 18M.2.hl.TZ0.6f: Find two elements in the same equivalence class which are inverses of each other with respect to...
- 18M.2.hl.TZ0.6e: Determine the equivalence classes into which R partitions S, giving the first four elements of...
- 18M.2.hl.TZ0.6d: Prove that the operation ∘ is not distributive over ∗.
- 18M.2.hl.TZ0.6c: State which elements of S are self-inverse with respect to ∗.
- 18M.2.hl.TZ0.6b: Show that every element of S has an inverse with respect to ∗.
- 18M.2.hl.TZ0.6a: Find the identity element of S with respect to ∗.
- 18M.1.hl.TZ0.9b.ii: Hence find the inverse function f−1.
- 18M.1.hl.TZ0.9b.i: Show that the function f is a bijection.
- 18M.1.hl.TZ0.9a: Given that A is the interval {x:0⩽x⩽3} and...
- 18M.1.hl.TZ0.7b.ii: Find the critical value for ˉx if she wants the probabilities of a Type I error and a...
- 18M.1.hl.TZ0.7b.i: She decides to use the same acceptance criteria as the previous investigator. Find the...
- 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.TZ0.4b.ii: Show that this group is cyclic.
- 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.TZ0.4a: Copy and complete the following Cayley table for the transformations of T1, T2, T3, T4, under the...
- 16M.2.hl.TZ0.6e: (i) Find the inverse of a+b√2. (ii) Hence show that a2−2b2...
- 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.TZ0.6c: Show that (i) 1−√2 has an inverse in J; (ii) 2+4√2 has no...
- 16M.2.hl.TZ0.6b: State the identity in J.
- 16M.2.hl.TZ0.8a: (i) Show that the order of Sn is n!; (ii) List the 6 elements of S3 in...
- 16M.2.hl.TZ0.8c: (i) Use mathematical induction to prove...
- 16M.2.hl.TZ0.8b: (i) Write down the matrices M1, M2 representing the permutations...
- 16M.1.hl.TZ0.1c: Determine whether or not f is (i) injective; (ii) surjective.
- 16M.1.hl.TZ0.1b: (i) State the value of f(1), giving a reason for your answer. (ii) Find...
- 16M.1.hl.TZ0.1a: Explain why only one of the following statements is true (i) 17⊂P; (ii) ...
- 16M.1.hl.TZ0.9b: The relation R is defined on Z+ by nRm if and only if...
- 16M.2.hl.TZ0.6a: Show that J is closed.
- 17M.2.hl.TZ0.8c.i: Show that the order of the element (n−1) is 2.
- 17M.2.hl.TZ0.8b: Show that Gn={Sn, ×n} is a group. You may assume that...
- 17M.2.hl.TZ0.8c.v: Determine the inverse of the element 3 in G31.
- 17M.2.hl.TZ0.8c.iv: Determine the inverse of the element 3 in G11.
- 17M.2.hl.TZ0.8c.iii: Explain why the inverse of the element 3 is 13(n+1) for some values of n but...
- 17M.2.hl.TZ0.8c.ii: Show that the inverse of the element 2 is 12(n+1).
- 17M.2.hl.TZ0.8a.ii: Show that, for...
- 17M.2.hl.TZ0.8a.i: Show that there are no elements a, b∈Sn such that a×nb=0.
- 17M.2.hl.TZ0.2b.ii: Determine the three equivalence classes.
- 17M.2.hl.TZ0.2b.i: Show that 3R10.
- 17M.2.hl.TZ0.2a: The group {G, ∗} has a subgroup {H, ∗}. The relation R...
- 17M.1.hl.TZ0.6c: The permutation group G is generated by P. Determine the element of G that is of...
- 17M.1.hl.TZ0.6b: Find P2.
- 17M.1.hl.TZ0.6a: Determine the order of P, justifying your answer.
- 17M.1.hl.TZ0.10b: Determine whether or not {H, ∗} is a subgroup of {G, ∗}.
- 17M.1.hl.TZ0.10a: Show that {G, ∗} is a group.
- 15M.2.hl.TZ0.9d: Show that if S is a subgroup of G, then f(S) is a subgroup of H.
- 15M.2.hl.TZ0.9c: Show that if G is Abelian, then H must also be Abelian.
- 15M.2.hl.TZ0.9b: Show that if x is an element of G, then f(x−1)=(f(x))−1.
- 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.TZ0.7b: A relation R is defined on S such that A is related to B if and only if there...
- 15M.1.hl.TZ0.15b: For each of ρ1 and ρ2 which is an equivalence relation, describe the...
- 15M.1.hl.TZ0.15a: For ρ1 and ρ2 determine whether or not each is reflexive, symmetric and...
- 15M.1.hl.TZ0.11: Prove that the function f:Z×Z→Z×Z...
- 15M.1.hl.TZ0.3d: Explain whether or not the group is cyclic.
- 15M.1.hl.TZ0.3c: State the order of each element.
- 15M.1.hl.TZ0.3b: Show that {S, +6} forms an Abelian group.
- 11M.1.hl.TZ0.3b: For a,b∈Z the relation aRb is defined if and only if ab=2k...
- 11M.2.hl.TZ0.6a: (i) Draw the Cayley table for the set S={0,1,2,3,4,5}...
- 11M.2.hl.TZ0.6b: Prove that a cyclic group with exactly one generator cannot have more than two elements.
- 11M.2.hl.TZ0.6c: H is a group and the function Φ:H→H is defined by Φ(a)=a−1 ,...
- 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,...
- 10M.2.hl.TZ0.1a: (i) Show that ∗ is associative. (ii) Find the identity element. (iii) Find...
- 10M.2.hl.TZ0.1b: The domain of ∗ is now reduced to S={0,2,3,4,5,6} and...
- 09M.1.hl.TZ0.1a: Show that R is an equivalence relation.
- 09M.1.hl.TZ0.1b: State the equivalence classes of R .
- 09M.2.hl.TZ0.2A.a: (i) Show that Z4 (the set of integers modulo 4) together with the operation...
- 09M.2.hl.TZ0.2A.b: Using Cayley tables or otherwise, show that G and...
- 09M.2.hl.TZ0.2B.b: the group is cyclic.
- 09M.2.hl.TZ0.2b: 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≠e , (i) show that b≠e ; (ii) show that G is not...
- 13M.2.hl.TZ0.5c: Determine (i) the equivalence class E containing 1+√2 ; (ii) the...
- 13M.2.hl.TZ0.5d: Show that (i) (1+√2)3∈F ; (ii) (1+√2)6∈E .
- 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 r1Rr2⇒r21Rr22 is...
- 13M.2.hl.TZ0.5e: Determine whether the set E forms a group under (i) the operation of addition; (ii)...
- 08M.1.hl.TZ0.2a: Draw the Cayley table for {G,∗} .
- 08M.1.hl.TZ0.2b: (i) Determine the order of each element of {G,∗}...
- 08M.1.hl.TZ0.2c: Solve the equation x∗6∗x=3 where x∈G .
- 08M.2.hl.TZ0.4A.a: The relation R1 is defined for a,b∈Z+ by aR1b if and only if...
- 08M.2.hl.TZ0.4B: Consider the group {G,∗} and let H be a subset of G defined...
- 08M.2.hl.TZ0.4B.b: The relation R2 is defined for a,b∈Z+ by aR2b if and only if...
- 07M.1.hl.TZ0.3a: Show that the set S of numbers of the form 2m×3n , where...
- 07M.1.hl.TZ0.3b: Show that {S,×} is isomorphic to the group of complex numbers...
- 07M.2.hl.TZ0.5a: The relation R is defined for x,y∈Z+ such that xRy if and only if...
- 07M.2.hl.TZ0.5b: Let S denote the set...
- 12M.1.hl.TZ0.1a: The set S1={2,4,6,8} and ×10 denotes...
- 12M.1.hl.TZ0.1b: Now consider the group {S1,×20} where...
- 12M.2.hl.TZ0.4A.a: Show that f is a bijection if A is non-singular.
- 12M.2.hl.TZ0.4A.b: Suppose now that A is singular. (i) Write down the relationship between...
- 12M.2.hl.TZ0.4B.a: Show that {S,+m} 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...
- 12M.2.hl.TZ0.4B.c: Suppose now that m=ab where a , b are unequal prime numbers. Show that...
- SPNone.1.hl.TZ0.6: The group {G,+} is defined by the operation of addition on the set...
- SPNone.1.hl.TZ0.14b: Show that S forms an Abelian group under matrix multiplication.
- SPNone.1.hl.TZ0.14c: Giving a reason, state whether or not this group is cyclic.
- SPNone.2.hl.TZ0.1a: Show that R is an equivalence relation.
- SPNone.2.hl.TZ0.1b: Determine the equivalence class containing (x1,y1) and interpret it geometrically.
- SPNone.2.hl.TZ0.4a: Copy and complete the following Cayley table.
- SPNone.2.hl.TZ0.4b: Show that {S,×9} is not a group.
- SPNone.2.hl.TZ0.4c: Prove that a group {G,×9} can be formed by removing two elements...
- SPNone.2.hl.TZ0.4d: (i) Find the order of all the elements of G . (ii) Write down all the proper...
- SPNone.2.hl.TZ0.4e: Solve the equation 4×9x×9x=1 .
- 14M.1.hl.TZ0.8: The group {G, ∗} has a subgroup {H, ∗}. The relation R...
- 14M.1.hl.TZ0.16: {G, ∗} is a group of order N and {H, ∗} is a proper...
- 14M.1.hl.TZ0.13: The function...
- 14M.2.hl.TZ0.2a: The set S contains the eighth roots of unity given by...
- 15M.1.hl.TZ0.3a: Construct the Cayley table for {S, +6}.
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⊂P; (ii) ...
- 17M.1.hl.TZ0.10b: Determine whether or not {H, ∗} is a subgroup of {G, ∗}.
- 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 ˉx if she wants the probabilities of a Type I error and a...
4.2
- 11M.1.hl.TZ0.3b: For a,b∈Z the relation aRb is defined if and only if ab=2k...
- 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 r1Rr2⇒r21Rr22 is...
- 13M.2.hl.TZ0.5c: Determine (i) the equivalence class E containing 1+√2 ; (ii) the...
- 13M.2.hl.TZ0.5d: Show that (i) (1+√2)3∈F ; (ii) (1+√2)6∈E .
- 08M.2.hl.TZ0.4A.a: The relation R1 is defined for a,b∈Z+ by aR1b if and only if...
- 08M.2.hl.TZ0.4B.b: The relation R2 is defined for a,b∈Z+ by aR2b if and only if...
- 07M.2.hl.TZ0.5a: The relation R is defined for x,y∈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 (x1,y1) and interpret it geometrically.
- 14M.1.hl.TZ0.8: The group {G, ∗} has a subgroup {H, ∗}. The relation R...
- 15M.1.hl.TZ0.15a: For ρ1 and ρ2 determine whether or not each is reflexive, symmetric and...
- 15M.1.hl.TZ0.15b: For each of ρ1 and ρ2 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 Z+ by nRm if and only if...
- 17M.2.hl.TZ0.2a: The group {G, ∗} has a subgroup {H, ∗}. The relation R...
- 17M.2.hl.TZ0.2b.i: Show that 3R10.
- 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 A is non-singular.
- 12M.2.hl.TZ0.4A.b: Suppose now that 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:Z×Z→Z×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 {x:0⩽x⩽3} 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 {G,∗} .
- SPNone.2.hl.TZ0.4a: Copy and complete the following Cayley table.
- 15M.1.hl.TZ0.3a: Construct the Cayley table for {S, +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 ∘ 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−√2 has an inverse in J; (ii) 2+4√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√2. (ii) Hence show that a2−2b2...
- 17M.2.hl.TZ0.8a.i: Show that there are no elements a, b∈Sn such that a×nb=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 12(n+1).
- 17M.2.hl.TZ0.8c.iii: Explain why the inverse of the element 3 is 13(n+1) for some values of n but...
- 17M.2.hl.TZ0.8c.iv: Determine the inverse of the element 3 in G11.
- 17M.2.hl.TZ0.8c.v: Determine the inverse of the element 3 in G31.
- 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={0,1,2,3,4,5}...
- 10M.2.hl.TZ0.1b: The domain of ∗ is now reduced to S={0,2,3,4,5,6} and...
- 09M.2.hl.TZ0.2A.a: (i) Show that Z4 (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≠e , (i) show that b≠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∈G .
- 07M.1.hl.TZ0.3a: Show that the set S of numbers of the form 2m×3n , where...
- 07M.2.hl.TZ0.5b: Let S denote the set...
- 12M.1.hl.TZ0.1a: The set S1={2,4,6,8} and ×10 denotes...
- SPNone.1.hl.TZ0.14b: Show that S forms an Abelian group under matrix multiplication.
- SPNone.2.hl.TZ0.4b: Show that {S,×9} is not a group.
- SPNone.2.hl.TZ0.4c: Prove that a group {G,×9} can be formed by removing two elements...
- SPNone.2.hl.TZ0.4e: Solve the equation 4×9x×9x=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, +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, ∗} is a group.
- 17M.2.hl.TZ0.8b: Show that Gn={Sn, ×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
None4.9
- 11M.2.hl.TZ0.6a: (i) Draw the Cayley table for the set S={0,1,2,3,4,5}...
- 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={0,2,3,4,5,6} and...
- 09M.2.hl.TZ0.2b: the group is cyclic.
- 08M.1.hl.TZ0.2b: (i) Determine the order of each element of {G,∗}...
- 12M.1.hl.TZ0.1a: The set S1={2,4,6,8} and ×10 denotes...
- 12M.2.hl.TZ0.4B.a: Show that {S,+m} 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 Sn is n!; (ii) List the 6 elements of S3 in...
- 16M.2.hl.TZ0.8b: (i) Write down the matrices M1, M2 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 P2.
- 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={0,1,2,3,4,5}...
- 10M.2.hl.TZ0.1b: The domain of ∗ is now reduced to S={0,2,3,4,5,6} and...
- 08M.1.hl.TZ0.2b: (i) Determine the order of each element of {G,∗}...
- 08M.2.hl.TZ0.4B: Consider the group {G,∗} 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, ∗} is a group of order N and {H, ∗} 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 Φ:H→H is defined by Φ(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 {S,×} is isomorphic to the group of complex numbers...
- 12M.1.hl.TZ0.1b: Now consider the group {S1,×20} where...
- SPNone.1.hl.TZ0.6: The group {G,+} 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)=(f(x))−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.