Maharashtra Board Class 12 Maths Part 1 Chapter 1 Mathematical Logic 1.10 Solutions

Question 1. Express the truth of each of the following statements by Venn diagrams:
(i) Some hardworking students are obedient.
Answer:
(i) Let U : set of all students
S : set of all hardworking students
O : set of all obedient students.
The shaded region represents the common elements between both sets. Then the Venn diagram representing the truth of the given statement is as below:

In simple words: The diagram shows two overlapping circles inside a box. The overlapping shaded part represents students who are both hardworking and obedient.

🎯 Exam Tip: Always define the universal set U and clearly label each circle with its corresponding set letter to secure full marks.

Question 1. Represent the following statements using Venn diagrams:
(ii) No circles are polygons.
(iii) All teachers are scholars and scholars are teachers.
(iv) If a quadrilateral is a rhombus, then it is a parallelogram.
Answer:
(ii) Let \( U \) : set of closed geometrical figures in the plane
\( P \) : set of all polygons
\( C \) : set of all circles.
Then the Venn diagram represents the truth of the given statement as follows:
\( P \cap C = \emptyset \)

(iii) Let \( U \) : set of all human beings
\( T \) : set of all teachers
\( S \) : set of all scholars.
Then the Venn diagram represents the truth of the given statement as below:
\( T = S \)

(iv) Let \( U \) : set of all quadrilaterals
\( R \) : set of all rhombuses
\( P \) : set of all parallelograms.
Then the Venn diagram represents the truth of the given statement as below:
\( R \subseteq P \)
Venn diagrams provide a clear visual representation of logical relationships between different sets.
In simple words: We use circles inside a rectangle to show how different groups of things relate to each other. Disjoint circles mean the groups have nothing in common, while one circle inside another means one group is completely part of the larger group.

🎯 Exam Tip: Always define the universal set \( U \) clearly at the beginning of your solution, and ensure the circles are properly labeled inside the rectangle to secure full marks.

Question 2. Draw the Venn diagrams for the truth of the following statements:
(i) Some share brokers are chartered accountants.
(ii) No wicket-keeper is a bowler in a cricket team.
Answer:
(i) Let \( U \) be the set of all human beings, \( S \) be the set of all share brokers, and \( C \) be the set of all chartered accountants. The statement "Some share brokers are chartered accountants" implies that there is at least one person who belongs to both categories. Therefore, the intersection of these two sets is non-empty, which is represented mathematically as \( S \cap C \neq \emptyset \). These diagrams visually simplify complex logical relationships between different groups of people.
(ii) Let \( U \) be the set of all human beings, \( W \) be the set of all wicket keepers, and \( B \) be the set of all bowlers. The statement "No wicket-keeper is a bowler in a cricket team" implies that there is no individual who is both a wicket-keeper and a bowler. Therefore, the two sets have no elements in common, which is represented mathematically as \( W \cap B = \emptyset \).
In simple words: To show "some" of two groups overlap, we draw two circles that cross each other and shade the middle part. To show "no" overlap between two groups, we draw two completely separate circles that do not touch.

🎯 Exam Tip: Always define the universal set \( U \) clearly before drawing Venn diagrams, and ensure you label each set with its corresponding capital letter to secure full marks.

Question 3. Represent the following statements by Venn diagrams:
(i) Some non-resident Indians are not rich.
(ii) No circle is a rectangle.
(iii) If n is a prime number and n ≠ 2, then it is odd.
Answer:
(i) Let U : set of all human beings
N : set of all non-resident Indians
R : set of all rich people.
Then the Venn diagram represents the truth of the given statement is as below: \( N - R \neq \phi \)

(ii) Let U : set of all geometrical figures
C : set of all circles
R : set of all rectangles
Then the Venn diagram represents the truth of the given statement is as below: \( C \cap R = \phi \)

(iii) Let U : set of all real numbers
P : set of all prime numbers n, where \( n \neq 2 \)
O : set of all odd numbers.
Then the Venn diagram represents the truth of the given statement is as below: \( P \subseteq O \)
In simple words: Venn diagrams help us see how different groups relate to each other. We can show if groups overlap slightly, are completely separate, or if one group fits entirely inside another.

🎯 Exam Tip: Clearly define the universal set \( U \) and all other sets before drawing. Label every set and the universal set \( U \) in your diagram to secure full marks.

Venn Diagram: \( P \subset O \)

The relation \( P \subset O \) indicates that set P is a proper subset of set O. In a Venn diagram, this is represented by drawing the circle for set P completely inside the circle for set O, within the universal set U.

🎯 Exam Tip: When representing the subset relation \( A \subset B \), always ensure the circle representing the subset (A) is drawn entirely inside the boundary of the parent set (B) to secure full marks.