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.