DP Mathematics HL Questionbank

• Syllabus

8.2

Sub sections and their related questions

Ordered pairs: the Cartesian product of two sets.

• 14M.3srg.hl.TZ0.3a: (i)     Sketch the set \(X \times Y\) in the Cartesian plane. (ii)     Sketch the set...
• 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...
• 16N.3srg.hl.TZ0.4a: Show that \(S\) is (i)     not reflexive; (ii)     symmetric; (iii)     transitive.
• 16N.3srg.hl.TZ0.4b: Explain why there exists an element \(a \in A\) that is not related to itself.
• 16N.3srg.hl.TZ0.4c: Hence prove that there is at least one element of \(A\) that is not related to any other element...
• 17N.3srg.hl.TZ0.3a: Show that R is an equivalence relation.
• 18M.3srg.hl.TZ0.3a.i: Show that \(R\) is reflexive.
• 18M.3srg.hl.TZ0.3a.ii: Show that \(R\) is symmetric.
• 18M.3srg.hl.TZ0.3b: Show, by means of an example, that \(R\) is not transitive.

Relations: equivalence relations; equivalence classes.

• 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...
• 12N.3srg.hl.TZ0.1a: Decide, giving a proof or a counter-example, whether \(xRy \Leftrightarrow x + y > 7\) is (i)...
• 12N.3srg.hl.TZ0.1b: 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.TZ0.1d: Decide, giving a proof or a counter-example, whether...
• 12N.3srg.hl.TZ0.1e: One of the relations from parts (a), (b), (c) and (d) is an equivalence relation. For this...
• 08M.3srg.hl.TZ1.4: (a)     Show that R is an equivalence relation. (b)     Describe, geometrically, the equivalence...
• 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...
• 11M.3srg.hl.TZ0.3a: Show that R is an equivalence relation.
• 11M.3srg.hl.TZ0.3b: Identify the three equivalence classes.
• 09M.3srg.hl.TZ0.3: The relation R is defined on \(\mathbb{Z} \times \mathbb{Z}\) such that...
• 09N.3srg.hl.TZ0.3: The relations R and S are defined on quadratic polynomials P of the...
• 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...
• 10N.3srg.hl.TZ0.1: Let R be a relation on the set \(\mathbb{Z}\) such that \(aRb \Leftrightarrow ab \geqslant 0\),...
• 13M.3srg.hl.TZ0.4a: Show that R is an equivalence relation.
• 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.
• 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...
• 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...
• 13N.3srg.hl.TZ0.4: Let \((H,{\text{ }} * {\text{)}}\) be a subgroup of the group \((G,{\text{ }} *...
• 14N.3srg.hl.TZ0.5c: Let \(\{ H,{\rm{ }} * {\rm{\} }}\) be a subgroup of \(\{ G,{\rm{ }} * {\rm{\} }}\). Let \(R\) be...
• 14N.3srg.hl.TZ0.5d: Let \(\{ H,{\rm{ }} * {\rm{\} }}\) be a subgroup of \(\{ G,{\rm{ }} * {\rm{\} }}\) .Let \(R\) be...
• 14N.3srg.hl.TZ0.5e: Let \(\{ H,{\rm{ }} * {\rm{\} }}\) be a subgroup of \(\{ G,{\rm{ }} * {\rm{\} }}\) .Let \(R\) be...
• 15M.3srg.hl.TZ0.3a: Show that the product of three consecutive integers is divisible by \(6\).
• 15M.3srg.hl.TZ0.3b: Hence prove that \(R\) is reflexive.
• 15M.3srg.hl.TZ0.3c: Find the set of all \(y\) for which \(5Ry\).
• 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.
• 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.TZ0.2c: The relation \(R\) is defined for \(a,{\text{ }}b \in \mathbb{R}\) so that \(aRb\) if and only if...
• 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...
• 16N.3srg.hl.TZ0.4a: Show that \(S\) is (i)     not reflexive; (ii)     symmetric; (iii)     transitive.
• 16N.3srg.hl.TZ0.4b: Explain why there exists an element \(a \in A\) that is not related to itself.
• 16N.3srg.hl.TZ0.4c: Hence prove that there is at least one element of \(A\) that is not related to any other element...
• 17N.3srg.hl.TZ0.3a: Show that R is an equivalence relation.
• 18M.3srg.hl.TZ0.3a.i: Show that \(R\) is reflexive.
• 18M.3srg.hl.TZ0.3a.ii: Show that \(R\) is symmetric.
• 18M.3srg.hl.TZ0.3b: Show, by means of an example, that \(R\) is not transitive.