Count members of overlapping groups using 'both'/'neither' reasoning, Venn diagrams, or the inclusion-exclusion principle (e.g., surveys of students who have cats and/or dogs).
Free, timed SHSAT practice sets on set logic, venn & inclusion-exclusion— every question includes a full worked explanation, and there's no account required.
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The key idea is inclusion-exclusion:
\(\text{at least one} = A + B - \text{both}\)
You subtract "both" because those people are counted once in A and again in B.
Example: of 60 students, 30 have a dog, 20 have a cat, and 8 have both.
If the question gives "neither" and asks for "both", work backward: at least one \(= 60 - \text{neither}\), then both \(= A + B - \text{at least one}\).
Drawing two overlapping circles and filling in the middle first makes every part clear. Common traps: adding A and B without subtracting the overlap, and giving "at least one" when the question asks for "neither".
Owners of at least one: \(20 + 23 - 6 = 37\). Neither: \(70 - 37 = 33\). (Not adding back the 6 who own both (\(70 - 20 - 23\)) gives 27. Subtracting only the dog owners (\(70 - 20\)) gives 50. Stopping at the number who own at least one gives 37.)
Take the hamster owners and remove those who own both: \(34 - 9 = 25\). (Not removing the 9 who own both gives 34. Finding the number who own a cat but not a hamster (\(29 - 9\)) gives 20. Counting everyone without a cat (\(74 - 29\)), which also counts those who own neither, gives 45.)
Owners of at least one: \(51 - 4 = 47\). Both: \(24 + 34 - 47 = 11\). (Forgetting the 4 who own neither (\(24 + 34 - 51\)) gives 7. Stopping at the number who own at least one gives 47. Stopping at the number who own only a cat (\(47 - 24\)) gives 23.)