Definitions: FSc Part 1 (Mathematics): PTB by Aurang Zaib

Definitions from Textbook of Algebra and Trigonometry Class XI, published by Punjab Textbook Board (PTB) Lahore, Pakistan. We are very thankful to Aurang Zaib for his valuable contribution.

A number which can be expressed in the form \( \dfrac{p}{q} \), where \( p, q \in \mathbb{Z} \) and \( q \neq 0 \), is termed as a rational number.

\( \dfrac{3}{4} \), \( \dfrac{7}{2} \)

A real number that cannot be represented as a fraction of two integers is called an irrational number.

\( \sqrt{2} \), \( \pi \)

The set comprising all rational and irrational numbers is referred to as the real numbers, denoted as \( \mathbb{R} \).

A decimal number that has a finite number of digits in its decimal part.

\( 0.25 \), \( 3.75 \)

A decimal in which one or more digits repeat indefinitely.

\( 0.3333... \), \( 1.234234... \)

A non-terminating decimal, also known as a non-recurring decimal, is a decimal representation of a number that neither terminates nor repeats. These decimals cannot be expressed as a fraction with integer numerator and denominator. They often represent irrational numbers.

\( \pi \) (pi) is a well-known non-terminating, non-recurring decimal. Its decimal representation is \( 3.1415... \). Similarly, \( \sqrt{2} \) is another example, with its decimal representation being \( 1.41421356... \).

A binary operation on a set \( A \) is a rule, typically denoted by \( \circ \) or \( \star \), that assigns to any pair of elements in \( A \) another element of \( A \).

In the set of real numbers \( \mathbb{R} \), two important binary operations are addition (+) and multiplication (\(\times\)). For example, for any real numbers \( a \) and \( b \), \( a + b \) and \( a \times b \) are also real numbers.

A complex number is a number of the form \( z = x + iy \), where \( x \) and \( y \) are real numbers, and \( i = \sqrt{-1} \) is the imaginary unit. Here, \( x \) is called the real part, and \( y \) is called the imaginary part of \( z \).

Some examples of complex numbers include \( 2 \), \( 3 + \sqrt{3}i \), and \( \dfrac{1}{2} + i \).

The real plane, also known as the coordinate plane, is the geometric plane where a coordinate system, typically consisting of horizontal and vertical axes, has been specified.

In the Cartesian coordinate system, the real plane consists of a horizontal x-axis and a vertical y-axis. Any point in this plane can be represented by an ordered pair \( (x, y) \), where \( x \) represents the horizontal position (abscissa) and \( y \) represents the vertical position (ordinate).

An Argand diagram is a graphical representation of complex numbers on the complex plane. It is similar to the Cartesian coordinate system, where the horizontal axis represents the real part, and the vertical axis represents the imaginary part of complex numbers.

Consider the complex number \( z = 3 + 2i \). In the Argand diagram, the real part \( x = 3 \) corresponds to the horizontal axis, and the imaginary part \( y = 2 \) corresponds to the vertical axis. Thus, \( z \) would be plotted as a point at coordinates \( (3, 2) \) on the Argand diagram.

The modulus of a complex number \( z = x + iy \) is the distance from the origin to the point representing the number on the complex plane. It is denoted by \( |z| \) or \( |(x, y)| \).

For the complex number \( z = 3 + 4i \), the modulus \( |z| \) is calculated as the square root of the sum of squares of its real and imaginary parts: \( |z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \). Therefore, \( |z| = 5 \).

A set is a well-defined collection of distinct objects or elements.

Consider the set of natural numbers: \( N = \{1, 2, 3, 4, \ldots\} \), where each element is distinct and well-defined.

There are 3 method to describe a set

Sets can be described using words.

$\mathbb{N}$ can be described descriptively as the set of all natural numbers.

Sets can be listed by enumerating their elements within braces.

$\mathbb{N}$ can be represented tabularly as \( N = \{1, 2, 3, 4, \ldots\} \).

In this method, a property common to all elements is described using set-builder notation.

Let \( A = \{x \ | \ x \) is any natural number\}. This implies that \( A \) consists of all natural numbers.

The number of elements in a set.

If \( A = \{3, 4\} \), then the order of \( A \) is 2.

Two sets are equal if they contain the same elements, regardless of the order.

\( A = \{2, 4, 6, 8\} \) and \( B = \{2, 8, 6, 4\} \) are equal sets since they have the same elements.

Two sets are equivalent if there exists a one-to-one correspondence between their elements.

\( A = \{2, 4, 6, 8\} \) and \( B = \{a, b, c, d\} \) are equivalent sets if each element in \( A \) corresponds to an element in \( B \).

A singleton set is a set that contains only one element.

• \( A = \{2\} \)
• \( B = \{5\} \)

A null set, also known as an empty set, is a set that contains no elements.

• \( \emptyset \)
• \( C = \{\} \)

A finite set is a set that contains a limited, countable number of elements.

• \( D = \{1, 2, 3, 4\} \)
• \( E = \{a, b, c\} \)

An infinite set is a set that contains an unlimited, uncountable number of elements.

• \( F = \{1, 2, 3, ...\} \) (Set of natural numbers)
• \( G = \{a, b, c, ...\} \) (Set of letters in the alphabet)

A subset is a set that contains elements from another set, where every element of the subset is also an element of the larger set.

• If \( A = \{1, 2\} \) and \( B = \{1, 2, 3\} \), then \( A \subseteq B \).

A proper subset is a subset that contains some, but not all, elements of another set.

• If \( C = \{1\} \) and \( D = \{1, 2\} \), then \( C \subset D \).

An improper subset is a subset that contains all elements of another set, including itself.

• If \( E = \{1, 2\} \), then \( E \subseteq E \).

The power set of a set is the set of all its subsets, including the empty set and itself.

• If \( A = \{1, 2\} \), then \( P(A) = \{\emptyset, \{1\}, \{2\}, \{1, 2\}\} \).

The universal set is the set containing all objects or elements under consideration.

• If we are considering the set of all integers, then the universal set could be denoted as \( \mathbb{Z} \).

The complement of a set relative to the universal set is the set of all elements not contained in the given set.

• If \( U = \{1, 2, 3, 4, 5\} \) and \( A = \{1, 2\} \), then \( A' = \{3, 4, 5\} \).

Deduction is the process of drawing logical conclusions from known facts or premises.

• If it is known that all men are mortal, and Socrates is a man, then it can be deduced that Socrates is mortal.

Induction is the process of drawing general conclusions based on specific instances or observations.

• If we observe that a coin is heads-up five times in a row, we might induce that it will always land heads-up, which may or may not be true.

Aristotelian logic is a deductive system in which every statement is regarded as either true or false.

• “All humans are mortal” would be regarded as either true or false in Aristotelian logic.

Non-Aristotelian logic is a deductive system that allows for more than two truth values, such as true, false, and unknown.

• In some systems of non-Aristotelian logic, a statement may be true, false, or neither true nor false.

A truth table is a table that shows all possible truth values of a given compound statement based on the truth values of its component parts.

Consider the truth table for the logical statement \( p \land q \):

\[ \begin{array}{|c|c|c|} \hline p & q & p \land q \\ \hline T & T & T \\ T & F & F \\ F & T & F \\ F & F & F \\ \hline \end{array} \]

A tautology is a statement that is true for all possible truth values of its variables.

\( p \rightarrow q \leftrightarrow (\neg q \rightarrow \neg p) \) is a tautology because its truth table shows that it is always true, regardless of the truth values of \( p \) and \( q \).

A contradiction is a statement that is always false, regardless of the truth values of its variables.

\( p \land \neg p \) is a contradiction because its truth table shows that it is always false.

A contingency is a statement that can be either true or false depending on the truth values of its variables.

\( (p \rightarrow q) \land (p \lor q) \) is a contingency because its truth table shows that it is true for some combinations of truth values of \( p \) and \( q \) and false for others.

A function is a relation between two non-empty sets \( A \) and \( B \), where each element of set \( A \) is related to exactly one element of set \( B \).

Let \( A = \{1, 2, 3\} \) and \( B = \{a, b, c\} \). A function \( f: A \rightarrow B \) could be defined as \( f(1) = a, f(2) = b, f(3) = c \).

A bijective function is a function that is both injective (one-to-one) and surjective (onto).

Consider the function \( f: \mathbb{R} \rightarrow \mathbb{R} \) defined as \( f(x) = 2x \). It is both injective and surjective, hence bijective.

An injective function is a function that assigns distinct elements of the domain to distinct elements of the codomain.

The function \( f: \{1, 2, 3\} \rightarrow \{a, b, c\} \) defined as \( f(1) = a, f(2) = b, f(3) = c \) is injective.

A groupoid is a non-empty set closed under a given binary operation.

Let \( G = \{a, b, c\} \) be a set with a binary operation \( * \). If for any elements \( x, y \) in \( G \), \( x * y \) is also in \( G \), then \( G \) is a groupoid under \( * \).

A binary operation on a set \( G \) is a mapping from the Cartesian product \( G \times G \) into \( G \). It associates every ordered pair of elements from \( G \) with a unique element of \( G \).

Addition \( + \) on the set of integers \( \mathbb{Z} \) is a binary operation because for any two integers \( a \) and \( b \), their sum \( a + b \) is also an integer.

A semi group is a non-empty set equipped with a binary operation that is associative.

The set of positive integers \( \mathbb{Z}^+ \) with the operation of multiplication \( \times \) forms a semi group because multiplication is associative for positive integers.

A monoid is a semi group with an identity element, i.e., a unique element that acts as an identity under the binary operation.

The set of non-negative integers \( \mathbb{Z}_0^+ \) with the operation of addition \( + \) forms a monoid because \( 0 \) serves as the identity element for addition.

A group is a non-empty set equipped with a binary operation that is associative, has an identity element, and every element has an inverse.

The set of integers \( \mathbb{Z} \) with the operation of addition \( + \) forms a group because for every integer \( a \), there exists its additive inverse \( -a \), and addition is associative.

An Abelian group (or commutative group) is a group where the binary operation is commutative.

The set of real numbers \( \mathbb{R} \) with the operation of addition \( + \) forms an Abelian group because addition of real numbers is commutative.

A linear function is a function of the form \( f(x) = mx + c \), where \( m \) and \( c \) are constants, and the graph of the function is a straight line.

The function \( f(x) = 2x + 3 \) is a linear function because its graph is a straight line with slope \( 2 \) and y-intercept \( 3 \).

A quadratic function is a function of the form \( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, and the highest power of the variable is \( 2 \).

The function \( f(x) = x^2 + 3x + 2 \) is a quadratic function because its highest power of the variable is \( 2 \).

A unary operation is an operation that takes a single input and produces a single output.

The square root function \( \sqrt{x} \) is a unary operation because it takes a single number \( x \) as input and produces its square root as output. For example, \( \sqrt{9} = 3 \).