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Q6:
Let U be the set of all triangles in a plane. If A is the set of all triangles with at least one angle different from 60°, what is A′?
Solution :
Step 1: Formal Definition of the Universal Set and Set A
Let $U$ be the set of all triangles in a plane. The complement of a set $A$ relative to the universal set $U$, denoted as $A'$, is defined as the set of all elements in $U$ that are not in $A$. Mathematically, $A' = \{x : x \in U \text{ and } x \notin A\}$. [Per the definition of the Complement of a Set].
The set $A$ is defined as:
$A = \{x \in U : \text{at least one angle of } x \neq 60^\circ\}$
Step 2: Logical Analysis of the Complement
To find $A'$, we must identify the condition that negates the definition of $A$. If a triangle $x$ is not in $A$, it must satisfy the negation of the predicate "at least one angle is different from $60^\circ$."
The negation of "at least one angle $\neq 60^\circ$" is "all angles $= 60^\circ$." [By the Laws of Logical Negation/De Morgan's Laws applied to quantifiers].
Step 3: Identification of the Set Geometry
Consider the properties of a triangle where all three interior angles are exactly $60^\circ$:
- The sum of angles in any triangle is $180^\circ$ [Euclidean Geometry, Angle Sum Property].
- If $\angle \alpha = \angle \beta = \angle \gamma = 60^\circ$, then $\alpha + \beta + \gamma = 60^\circ + 60^\circ + 60^\circ = 180^\circ$.
- A triangle with three equal angles is, by definition, an equilateral triangle.
Step 4: Formulation of the Resulting Set
Since $A'$ contains all triangles in $U$ that are not in $A$, and $A$ consists of all triangles that are not equilateral (as having at least one angle $\neq 60^\circ$ implies the triangle is not equiangular), $A'$ must contain only those triangles that are equilateral.
| Set | Condition |
|---|---|
| $A$ | $\exists \theta \in \{\text{angles of } x\} : \theta \neq 60^\circ$ |
| $A'$ | $\forall \theta \in \{\text{angles of } x\} : \theta = 60^\circ$ |
Therefore, $A' = \{x \in U : x \text{ is an equilateral triangle}\}$.
Final Solution: A' is the set of all equilateral triangles in the plane.
More Questions from Class 11 Mathematics Sets EXERCISE 1.5
- Q1(i): Let $U = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$, $A = \{ 1, 2, 3, 4\}$, $B = \{ 2, 4, 6, 8 \}$ and $C = \{ 3, 4, 5, 6 \}$. Find (i) A′
- Q1(ii): Let $U = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$, $A = \{ 1, 2, 3, 4\}$, $B = \{ 2, 4, 6, 8 \}$ and $C = \{ 3, 4, 5, 6 \}$. Find (ii) B′
- Q1(iii): Let $U = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$, $A = \{ 1, 2, 3, 4\}$, $B = \{ 2, 4, 6, 8 \}$ and $C = \{ 3, 4, 5, 6 \}$. Find (iii) $(A \cup C)′$
- Q1(iv): Let $U = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$, $A = \{ 1, 2, 3, 4\}$, $B = \{ 2, 4, 6, 8 \}$ and $C = \{ 3, 4, 5, 6 \}$. Find (iv) $(A \cup B)′$
- Q1(v): Let $U = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$, $A = \{ 1, 2, 3, 4\}$, $B = \{ 2, 4, 6, 8 \}$ and $C = \{ 3, 4, 5, 6 \}$. Find (v) $(A′)′$
- Q1(vi): Let $U = \{ 1, 2, 3, 4, 5, 6, 7, 8, 9 \}$, $A = \{ 1, 2, 3, 4\}$, $B = \{ 2, 4, 6, 8 \}$ and $C = \{ 3, 4, 5, 6 \}$. Find (vi) $(B – C)′$
- Q2(i): If $U = \{ a, b, c, d, e, f, g, h\}$, find the complements of the following sets : (i) $A = \{a, b, c\}$
- Q2(ii): If $U = \{ a, b, c, d, e, f, g, h\}$, find the complements of the following sets : (ii) $B = \{d, e, f, g\}$
- Q2(iii): If $U = \{ a, b, c, d, e, f, g, h\}$, find the complements of the following sets : (iii) $C = \{a, c, e, g\}$
- Q2(iv): If $U = \{ a, b, c, d, e, f, g, h\}$, find the complements of the following sets : (iv) $D = \{ f, g, h, a\}$
- Q3(i): Taking the set of natural numbers as the universal set, write down the complements of the following sets: (i) $\{x : x$ is an even natural number\}
- Q3(ii): Taking the set of natural numbers as the universal set, write down the complements of the following sets: (ii) $\{ x : x$ is an odd natural number \}
- Q3(iii): Taking the set of natural numbers as the universal set, write down the complements of the following sets: (iii) $\{x : x$ is a positive multiple of 3\}
- Q3(iv): Taking the set of natural numbers as the universal set, write down the complements of the following sets: (iv) $\{ x : x$ is a prime number \}
- Q3(ix): Taking the set of natural numbers as the universal set, write down the complements of the following sets: (ix) $\{ x : 2x + 5 = 9\}$
- Q3(v): Taking the set of natural numbers as the universal set, write down the complements of the following sets: (v) $\{x : x$ is a natural number divisible by 3 and 5\}
- Q3(vi): Taking the set of natural numbers as the universal set, write down the complements of the following sets: (vi) $\{ x : x$ is a perfect square \}
- Q3(vii): Taking the set of natural numbers as the universal set, write down the complements of the following sets: (vii) $\{ x : x$ is a perfect cube\}
- Q3(viii): Taking the set of natural numbers as the universal set, write down the complements of the following sets: (viii) $\{ x : x + 5 = 8 \}$
- Q3(x): Taking the set of natural numbers as the universal set, write down the complements of the following sets: (x) $\{ x : x \ge 7 \}$
- Q3(xi): Taking the set of natural numbers as the universal set, write down the complements of the following sets: (xi) $\{ x : x \in N$ and $2x + 1 > 10 \}$
- Q4(i): If $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9 \}$, $A = \{2, 4, 6, 8\}$ and $B = \{ 2, 3, 5, 7\}$. Verify that (i) $(A \cup B)′ = A′ \cap B′$
- Q4(ii): If $U = \{1, 2, 3, 4, 5, 6, 7, 8, 9 \}$, $A = \{2, 4, 6, 8\}$ and $B = \{ 2, 3, 5, 7\}$. Verify that (ii) $(A \cap B)′ = A′ \cup B′$
- Q5(i): Draw appropriate Venn diagram for each of the following : (i) $(A \cup B)′$
- Q5(ii): Draw appropriate Venn diagram for each of the following : (ii) $A′ \cap B′$
- Q5(iii): Draw appropriate Venn diagram for each of the following : (iii) $(A \cap B)′$
- Q5(iv): Draw appropriate Venn diagram for each of the following : (iv) $A′ \cup B′$
- Q7(i): Fill in the blanks to make each of the following a true statement : (i) $A \cup A′ = . . .$
- Q7(ii): Fill in the blanks to make each of the following a true statement : (ii) $\phi′ \cap A = . . .$
- Q7(iii): Fill in the blanks to make each of the following a true statement : (iii) $A \cap A′ = . . .$
- Q7(iv): Fill in the blanks to make each of the following a true statement : (iv) $U′ \cap A = . . .$
CBSE Solutions for Class 11 Mathematics Sets
Chapters in CBSE - Class 11 Mathematics
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