Pariksha SaathiSet Theory Questions with Solutions
Most Set Theory questions here reduce to one formula — the union of two groups equals their sizes added together, minus whatever they share — since adding the overlap twice is the error this formula exists to fix. Here’s the concept, four fully worked examples, and a few to try yourself.
The concept, quickly
- For two sets: |A ∪ B| = |A| + |B| − |A ∩ B| — adding the two group sizes directly double-counts whoever is in both, so the overlap is subtracted back out once.
- "At least one" means the union, |A ∪ B|. "Neither" means everyone outside the union: Total − |A ∪ B|.
- "Only A" (in A but not B) = |A| − |A ∩ B|. "Exactly one of A or B" = |A ∪ B| − |A ∩ B| (the union, minus whoever is in both).
- Draw two overlapping circles and fill in the overlap number first — the "only A" and "only B" regions are then whatever’s left in each circle after removing the overlap.
Worked examples
1. In a class of 60 students, 35 play cricket and 25 play football. If 10 students play both, how many play at least one of the two games?
Solution
- |Cricket ∪ Football| = 35 + 25 − 10 = 50.
Answer: 50
2. In a survey of 100 people, 60 like tea and 50 like coffee. If 20 like both, how many like neither?
Solution
- |Tea ∪ Coffee| = 60 + 50 − 20 = 90.
- Neither = Total − union = 100 − 90 = 10.
Answer: 10
3. In a group of 50 students, 30 study Math, 25 study Physics, and 15 study both. How many study only Math (not Physics)?
Solution
- Only Math = |Math| − |Math ∩ Physics| = 30 − 15 = 15.
Answer: 15
4. In a class of 80 students, 50 play cricket, 40 play football, and 20 play both. How many play exactly one of the two games?
Solution
- |Cricket ∪ Football| = 50 + 40 − 20 = 70.
- Exactly one = union − both = 70 − 20 = 50.
Answer: 50
Try these yourself
Find the union first, then adjust for "neither," "only," or "exactly one" as needed, then check against the answer.
1. In a class of 70 students, 40 play chess and 30 play carrom. If 15 play both, how many play at least one game?
Answer: 55
2. In a survey of 120 people, 70 like movies and 60 like music. If 25 like both, how many like neither?
Answer: 15
3. In a group of 90 students, 55 study English, 45 study Hindi, and 20 study both. How many study only English?
Answer: 35
Where this comes up
Set Theory is tested directly in the Quantitative/Numerical Aptitude section of every exam Pariksha Saathi covers — SSC CGL, SSC MTS, SSC CHSL, IBPS PO, IBPS Clerk, SBI PO and SBI Clerk — most often in SSC CGL.
Practice more Set Theory questions
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