courses › Counting & Discrete Math

Overlapping groups, subsets, truth tables, and “if … then”.

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• Overlapping groups par 18 s not tried yet

  worked example

  Of 47 gym members, 12 swim, 23 run, and 2 do both. How many swim or run (or both)?

  Answer: 33

  1. Adding 12 and 23 counts the 2 who do both twice, so take them off once:
  2. 12 + 23 − 2 = 33.

• Counting subsets par 11 s not tried yet

  worked example

  How many non-empty subsets does a set with 4 elements have?

  Answer: 15

  1. Each element is in or out: 2⁴ = 16 subsets.
  2. Leave out the empty one: 16 − 1 = 15.

• “If … then” par 17 s not tried yet

  worked example

  Which statement means the same as “If a number is divisible by 4, it is even”?

  1. If a number is even, it is divisible by 4
  2. A number is divisible by 4 and it is even
  3. If a number is not even, it is not divisible by 4
  4. If a number is not divisible by 4, it is not even

  Answer: If a number is not even, it is not divisible by 4

  1. Swapping the two parts and negating both (the contrapositive) keeps the meaning: “If a number is not even, it is not divisible by 4”.
  2. The converse and the inverse can be false while the original is true: 6 is even but not divisible by 4.

• Truth-table rows par 20 s not tried yet

  worked example

  How many of the 8 rows of the truth table make (¬p ∨ q) → r true? Key: ∨ = or, ¬ = not, → = if … then.

  Answer: 5

  1. Check all 8 rows, written as (p, q, r) with T for true and F for false.
  2. False only in TTF, FTF and FFF.
  3. So 5 of the 8 rows make it true.

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