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SECTION B (Subjective Type Questions)

A. Short answer type questions.
1. Write the set for B = {x: x is a natural number less than 8}.
Ans. B = {1,2,3,4,5,6,7}.
2. Let U = {2, 3, 4, 5, 6}, A = {2, 3} and B = {3, 4, 5}.
Find A′, B′, A′ B′ and A B.

Ans. U = {2, 3, 4, 5, 6},
A = {2, 3} and B = {3, 4, 5}
A′ = {4, 5, 6} B′ = {2, 6}

Hence, A′ B′ = {6} Also, A B = {2, 3, 4, 5}.
3. Identify and briefly explain the following relational operators:
a. σ b. π c. d.
Ans. ● Selection: This is a unary operation that selects only some of the tuples (records) of the relation. The
selection operation yields a subset of the tuples (records) from a relation (table) that satisfies a given
condition.
● Projection: The projection of a relation is defined as a projection of all its tuples over some set of
attributes. This operation yields a vertical subset of the relation (table).
● Intersection: The intersection operation is a binary operation. The intersection operation yields the
common tuples from the two tables.
● Union: The union operation combines two or more relations (tables). The resultant relation of the
union operation contains tuples that are in either of the relation or in both relations.
4. Consider the following matrix A:

 2 5 19 − 17
 5 
 35 − 2 2 12 
 
 3 1 − 5 17 
a. Find the order of the matrix.

b. Write the elements of a , a , a , a , and a .
13
22
24
23
33
Ans. a. Order of the matrix is 3 × 4.
5
b. 19, –2, –5, 12, .
2
5. Explain the relationship between variance and mean of a data set.
Ans. The variance represents how far the data in your sample are grouped around the mean. Data sets
with low variance have data grouped closely about the mean. Data sets with high variance have data
grouped far from the mean.
6. How are vectors used to represent data in machine learning?
Ans. Each data point is represented as a vector of numerical features. For example, a person’s height, weight,
and age may be stored as a vector (height, weight, age).
7. Define the intersection of two sets.
Ans. The intersection of two sets includes only those elements that are common to both sets.

Mathematics for AI 143