Set Theory & Algebra MCQ Quiz - Objective Question with Answer for Set Theory & Algebra - Download Free PDF

→ LUB of (9, 1) = 9

∴ 1 cannot be its complement

→ LUB of (9, 2) = 36

GLB of (9, 2) = 1

∴ 2 is its complement

→ LUB of (9, 3) = 9

∴ 3 cannot be its complement

→ LUB of (9, 4) = 36

GLB of (9, 4) = 1

∴ 4 is its complement

→ LUB of (9, 6) = 36

GUB of (9, 6) = 3

∴ 6 cannot be its complement

→ LUB of (9, 36) = 36

GUB of (9, 36) = 9

∴ 36 cannot be its complement

Complement 9 are: 2 and 4

Important Points:

GLB is greatest lower bound

A half adder circuit is basically made up of an a AND gate with XOR gate as shown below.

• A half adder is also known as XOR gate because XOR is applied to both inputs to produce the sum
• Half adder can add only two bits (A and B) and has nothing to do with the carry
• If the input to a half adder has a carry, then it will neglect it and adds only the A and B bits
• That means the binary addition process is not complete and that's why it is called a half adder

Sum (S) = A⊕B, Carry = A.B

The Correct Answer is I but not II.

• NAND gate is a functionally complete set of gates.
• In the logic gate, a functionally complete collection of logical connectives or Boolean operators is one which can be used to express all possible truth tables by combining members of the set into a Boolean expression.
• A well-known complete set of connectors is {AND, NOT} and each of the singleton sets {NAND} is functionally complete, consisting of binary conjunction and negation.
• A NAND gate is a logic gate that generates a false output only if all its inputs are valid, so its output is complementary to that of an AND gate.
• A low output only results if all the inputs to the gate are high; a high output results if any input is low.

Key Points

Explanation:
The question is about non-null sets A, B, C \(\Rightarrow 4^3\) – (Any set is Empty)
Consider \(A = \phi\)
Universal set \(= \{2,3,5\}\) contains 3 elements \(\Rightarrow~ B~ has~ 2^3\) possible choices, and for each possible B set, we need to calculate possible sets of C. As \(B \subseteq C\), the elements present in the B, should be present in C. Remaining elements of Universal set has two choices; present in C or not present in C.

\(if ~B = \phi\) (number of elements in B = 0) \(\Rightarrow\) number of possible sets for \(C = 2^{n-0} = 2^3\)
\( \textbf{if B = \{1\}}\) (number of elements in B = 1) \(\Rightarrow\) number of possible sets for \(C = 2^{n-1} = 2^2\)
\( \textbf{if B = \{2\}}\) (number of elements in B = 1) \(\Rightarrow\) number of possible sets for \(C = 2^{n-1} = 2^2\)
\( \textbf{if B = \{3\}}\) (number of elements in B = 1) \(\Rightarrow\) number of possible sets for \(C = 2^{n-1} = 2^2\)
\( \textbf{if B = \{1, 2\}}\) (number of elements in B = 2) \(\Rightarrow\) number of possible sets for \(C = 2^{n-2} = 2^1\)
\( \textbf{if B = \{1, 3\}}\) (number of elements in B = 2) \(\Rightarrow\) number of possible sets for \(C = 2^{n-2} = 2^1\)

\(\textbf{if } B = \{2, 3\}\) (number of elements in B = 2) \(\Rightarrow\) number of possible sets for \(C = 2^{n-2} = 2^1\)
\(​\textbf{if } B = \{1, 2, 3\}\) (number of elements in B = 3)} \(\Rightarrow\) number of possible sets for \(C = 2^{n-3} = 2^0\)
∴ \( \binom{n}{0} \cdot 2^n + \binom{n}{1} \cdot 2^{n-1} + \cdots + \binom{n}{r} \cdot 2^{n-r} \cdots + \binom{n}{n} \cdot 2^0 = (1 + 2)^n = 3^n = 3^3 = 27 \)
\(∴ \text{when } A = \phi, \text{ possible sets of B and C are 27} \)
Final answer = \(4^3 - 3^3 = 37\)

Concept use:

For Finding the Range of f(x): Convert the Function in the Form of Y then find out the domain of that function 

Calculation:
\(√{9-x^{2}}\) = f(x) (given)

\(√{9-x^{2}}\) = y 

Squaring both sides 

y² = 9 - x²

x² = 9 - y²

x = √9 - y²

We can put y from -3 to 3 

From the Given options [0, 3] Satisfied the Value of y 

Answer:19 to 19

Data

f: R -> R 

continuous in [-3.3]

differentiable in (-3,3) 

f(-3) = 7

 f'(x) < = 2

Calculation:

=>f'(x) < = 2

Integrating both side from -3 to 3

 \(\mathop \smallint \limits_{ - 3}^3 f'\left( x \right)dx <= 2\mathop \smallint \limits_{ - 3}^31dx\)

[f(3)-f(-3)] <= 2[3-(-3)]

f(3)<= 7 + 2(6)

f(3)<=19

Data:

Number of elements in a set = n = 5

Formula:

Total number of reflexive relations in a set = \(2^{n^2 -n}\)2n2−n2n2−n" role="presentation" style="display: inline; position: relative;" tabindex="0">-1--12n2−n2n2−n" id="MathJax-Element-4-Frame" role="presentation" style="position: relative;" tabindex="0">2n2−n2n2−n 2n2−n

Calculation:

Total number of reflexive relations in a set =  \(2^{5^2 -5} =2^{20}\)242−4=212" role="presentation" style="display: inline; position: relative;" tabindex="0">SSo, the

So, the correct answer is 2²⁰242−4=212" role="presentation" style="display: inline; position: relative;" tabindex="0">42−4

Concept: 

Important Axioms and De Morgan's laws of Boolean Algebra:

1. Double inversion \(\overline{\overline A} = A\)
2. A . A = A
3. A . \(\overline A \)  = 0
4. A + 1 = 1
5. A + A = A
6. A + \(\overline A \)  = 1

De Morgan's laws:

Law 1: \(\overline {{\bf{A}} + {\bf{B}}} = \overline{A}\;.\overline B\)

Law 2: \(\overline {{\bf{A}}\;.{\bf{B}}} = \overline A +\overline B\)

Calculation:

Let the given function be Y

Y = AB + AC

Now expanding by using the important properties of boolean algebra:

Y = AB(C + C̅) + AC(B + B̅)

Y = ABC + ABC̅ + ACB + ACB̅ 

As  ABC + ACB = ABC 

Y = ABC + ABC̅ + ACB̅ 

Y can also be written as:

Y = ABC + ABC' + AB'C

Hence option (4) is the correct answer.

Important Points

General points:

• sin(x) is the function having a range between -1 and +1.
• Log(x) is defined only when x is positive and greater than zero.

log (sin(x)) is defined only when 0 < sin(x) ≤ 1, and then range will be (−∞,0]

So, log [log (sin(x))] is undefined as the logarithm of non-positive numbers isn’t defined for real numbers.

Hence, Domain of log [ log (sin(x))]: ∅ (empty set)

Also, Range of log [log (sin(x))]: ∅ (empty set)

Concept -

In set theory, the intersection and union are basic operations.

Union of two sets is defined as a set that contains all the values that occur in both sets without repetition.

The intersection of two sets is defined as a set that contains the values which are common to both sets.

Explanation -

A∩B = {3, 4}, A∩C = {4, 6, 8}

(A∩B)∪(A∩C) = {3, 4}∪{4, 6, 8}

= {3, 4, 6, 8}

Hence option(iii) is correct.

Concept

According to the Lagrange theorem order of subgroups must divide the order of the group.

Property of group says if a group has prime order then it is cyclic.

Explanation:

Since the order of G is 6. Therefore  its subgroup may have order 1,2,3,6

H is one of its subgroups with condition 1< |H| <6 so H may be of order 2 or 3  which is prime 

Hence H must be cyclic

The order of G is 6 which is not prime and hence it may or may not  be cyclic 

Therefore option 2 is correct

Explanation - 

Symmetric difference of two sets is set which contains elements which are in exactly one set.

\(A\Delta B = (A-B) \cup (B-A)\)

= {2, 4, 9}

hence option (ii) is true.

When two switches are connected in parallel, then the circuit acts as an OR gate

The input X is connected to output Y when at least one of the switch A and Switch B is closed.

From the above truth table, the circuit diagram represents an OR gate i.e. (A + B)

When two switches are connected in series, then the circuit acts as an AND gate.

Now, the input X is connected to output Y when both the switch A and Switch B are closed.

From the above truth table, the circuit diagram represents an AND gate i.e. Y = AB

Concept:  

• A relation R in a set A is called
•   Reflexive, if (a, a) є R, for every a є A,

For example, a relation that implies a line is parallel to another line is a reflexive relation since each line is parallel to itself.

• Symmetric, if (a₁, a₂) є R implies that (a₂, a₁) є R for all a_1, a₂ є A,

For example, a relation that implies a line is parallel to another line is a symmetric relation. For example, if line 1 is parallel to line 2, then line 2 is parallel to line 1.

• Transitive, if (a₁, a₂) є R and (a₂, a₃) є R implies that (a₁, a₃) є R for all a_1, a_2, a₃ є A.

If a relation R is reflexive, symmetric, and transitive at once, then it is said to be an equivalence relation.

Calculation:

Given:

Relation R is defined by function R = {(a, b) | (a - b) = km for some fixed integer m and a, b, k ∈ z}.

1) For the same number a, R = a - a = 0 × m, where m is a fixed integer and 0 є z, hence the relation R is reflexive.

2) For two numbers (a, b) if R = a - b = km, where m is a fixed integer and k є z, then for (b, a), R = b - a = -(a - b) = -km, where -k є z. Hence relation R is symmetric.

3) Consider three numbers a, b, and c. If, for (a, b), R = a - b = km and for (b, c), R = b - c = lm, where m is a fixed integer and k, l є z, then for (a, c), R = a - c = a - b + b - c = km + lm = m(k + l), where (k + l) є z, since k, l є z. Hence relation R is transitive.

Hence, relation R is an equivalence relation.

Hence, the correct answer is option 4.

A half adder circuit is basically made up of an a AND gate with XOR gate as shown below.

• A half adder is also known as XOR gate because XOR is applied to both inputs to produce the sum
• Half adder can add only two bits (A and B) and has nothing to do with the carry
• If the input to a half adder has a carry, then it will neglect it and adds only the A and B bits
• That means the binary addition process is not complete and that's why it is called a half adder

Sum (S) = A⊕B, Carry = A.B