DP Mathematics HL Questionbank

• Syllabus

8.6

Sub sections and their related questions

The identity element $e$.

• 12N.3srg.hl.TZ0.4b: State the identity element for G under $*$.
• SPNone.3srg.hl.TZ0.2d: $\odot$ has an identity element.
• 13M.3srg.hl.TZ0.1d: has an identity element.
• 14M.3srg.hl.TZ0.1: The binary operation $\Delta$ is defined on the set $S =$ {1, 2, 3, 4, 5} by the following...
• 15M.3srg.hl.TZ0.2a: Find the element $e$ such that $e * y = y$, for all $y \in S$.
• 15M.3srg.hl.TZ0.2c: Determine whether or not $e$ is an identity element.
• 15N.3srg.hl.TZ0.3b: State the identity element in $\{ G,{\text{ }} \circ \}$.
• 17N.3srg.hl.TZ0.4c: Show that 2 is the identity element.
• 18M.3srg.hl.TZ0.5a: Prove that $f \circ f$ is the identity function.

The inverse ${a^{ - 1}}$ of an element $a$.

• 12N.3srg.hl.TZ0.4c: For $x \in G$ find an expression for ${x^{ - 1}}$ (the inverse of x under $*$).
• 14N.3srg.hl.TZ0.5b: Verify that the inverse of $a * {b^{ - 1}}$ is equal to $b * {a^{ - 1}}$.  
• 15M.3srg.hl.TZ0.1b: (i)     State the inverse of each element. (ii)     Determine the order of each element.
• 15N.3srg.hl.TZ0.3c: Find (i)     $p \circ p$; (ii)     the inverse of $p \circ p$.
• 17N.3srg.hl.TZ0.4c: Show that 2 is the identity element.
• 18M.3srg.hl.TZ0.5a: Prove that $f \circ f$ is the identity function.

Proof that left-cancellation and right-cancellation by an element a hold, provided that a has an inverse.

• 11M.3srg.hl.TZ0.5a: Given that p , q and r are elements of a group, prove the left-cancellation rule, i.e....
• 17N.3srg.hl.TZ0.4c: Show that 2 is the identity element.
• 18M.3srg.hl.TZ0.5a: Prove that $f \circ f$ is the identity function.

Proofs of the uniqueness of the identity and inverse elements.

• 17N.3srg.hl.TZ0.4c: Show that 2 is the identity element.
• 18M.3srg.hl.TZ0.5a: Prove that $f \circ f$ is the identity function.