5.5 Workshop 5 (Reflection)

In this workshop, you’ll receive feedback from your tutor on the proof you submitted for this block. Your tutor will give your group general feedback on the key areas for improvement and commendation that they noticed whilst marking your group’s proofs and they will also have a one-to-one conversation with you about your specific feedback. It is important that you understand the written feedback your tutor has given you to improve, especially if your proof has not been awarded a pass yet. So make sure you take this opportunity to ask about any part of the feedback you’re unsure of.

When you’re not discussing feedback with your tutor, work with your group on the summary task below. This task has been chosen to draw on a variety of concepts you’ve been exploring throughout this block and is deliberately open ended to give you more practice at forming conjectures of your own.

Task 5.21.

Is the set ℚ\mathbb{Q} countable or uncountable? What about the set ℝ∖ℚ\mathbb{R}\setminus\mathbb{Q}? Prove any claims that you make.

Extension Task

Definition (Comparing Cardinalities).

If AA and BB are sets that do NOT have the same cardinality, but there exists an injective function f:A→Bf:A\rightarrow B, then we write |A|<|B||A|<|B| and say that AA has cardinality strictly smaller than the cardinality of BB.

Task 5.22 (EXTENSION).

Compare the cardinality of a set with the cardinality of its power set.

  1. 1.
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    Do these sets have the same cardinality? If not, which has larger cardinality? What if we’re dealing with infinite sets? Prove any claims that you make.

    Hint: You may find a diagonal argument, like the one used to prove that (0,1)(0,1) is uncountable at the end of the B3 summary notes, helpful.

  2. 2.
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    Can you use what you’ve found in this extension task to find different sizes of infinity? How many different sizes of infinity are there?