☰ Contents · Mathematics

Correct reasoning and problem solving

Lessons 6–7 · 2 lessons · M. A. Mirzaahmedov, Sh. N. Ismailov, A. Q. Amanov (algebra and analysis), B. Q. Haydarov (geometry). Mathematics Grade 10, Parts 1 and 2, 1st edition. EXTREMUM PRESS, Tashkent, 2017
7

Problem solving: sets and logic

Textbook, Part 1: pp. 38–47
GoalSolve problems on two and three sets with Venn diagrams and the add–subtract formula; translate text arguments into the language of logic.
New words
Venn diagram · Venn diagrammasiintersection · kesishmanumber of elements · elementlar sonilogical form · mantiqiy shakl
Explanation

Start a problem on sets with a Venn diagram: name each region (only A, both A and B, only B, neither) and write equations from the data. For two sets n(A ∪ B) = n(A) + n(B) − n(A ∩ B), and the number of elements in neither set is n(U) − n(A ∪ B). For three sets n(A ∪ B ∪ C) = n(A) + n(B) + n(C) − n(A ∩ B) − n(A ∩ C) − n(B ∩ C) + n(A ∩ B ∩ C); the “only A” region equals n(A) − n(A ∩ B) − n(A ∩ C) + n(A ∩ B ∩ C). It is convenient to fill in the innermost region — those in all three sets — first. To solve a text argument, first name the simple statements p, q, r, write the logical form, and then check it by a truth table or by a known form (modus ponens, modus tollens and others).

Worked examples
Of 32 students, 20 like chess, 15 like swimming and 6 like neither. Those liking at least one: 32 − 6 = 26. Those liking both: 20 + 15 − 26 = 9.
60 students: 30 in the maths club, 25 in physics, 20 in chemistry; 10 in maths and physics, 8 in maths and chemistry, 6 in physics and chemistry, 3 in all three. Union: 30 + 25 + 20 − 10 − 8 − 6 + 3 = 54, so 60 − 54 = 6 are in no club. Only maths: 30 − 10 − 8 + 3 = 15.
Class activity

Class survey: each student answers three yes/no questions (for example bicycle, football, chess). The data are placed on a Venn diagram and the number of students in each of the eight regions is found.

Practice
1
Write the formula for the number of elements in the union of three sets.
2
n(U) = 45, n(A) = 20, n(B) = 18, n(A ∩ B) = 7. How many elements lie in neither A nor B?
3
Each of 45 library visitors took a fiction or a science book; 31 took fiction and 22 took science. How many took both kinds?
4
“If the bus is late, Nodira is late for class. Nodira was not late. So the bus was not late.” Write its form and justify its correctness.