FSc Pre-Engineering Mathematics Syllabus

Define the difference $A \setminus B$ and the complement $A'$ of a set.

Difference: $A \setminus B = \{x : x \in A \text{ and } x \notin B\}$. Complement: $A' = U \setminus A$, the elements of the universal set $U$ not in $A$.

If a set has $n$ elements, how many subsets does it have?

It has $2^{n}$ subsets (including $\varnothing$ and the set itself).

State De Morgan's laws for sets.

$(A \cup B)' = A' \cap B'$ and $(A \cap B)' = A' \cup B'$.

State the distributive laws of algebra of sets.

$A \cup (B \cap C) = (A \cup B) \cap (A \cup C)$ and $A \cap (B \cup C) = (A \cap B) \cup (A \cap C)$.

State the commutative and associative laws for set union and intersection.

Commutative: $A \cup B = B \cup A$, $A \cap B = B \cap A$. Associative: $(A \cup B) \cup C = A \cup (B \cup C)$, $(A \cap B) \cap C = A \cap (B \cap C)$.

What does a Venn diagram represent, and how are the universal set and other sets shown?

A Venn diagram visually represents sets and their relationships. The universal set $U$ is drawn as a rectangle, and individual sets are drawn as circles (or regions) inside it; overlaps show intersections.

Define the Cartesian product $A \times B$ of two sets.

$A \times B = \{(a,b) : a \in A,\ b \in B\}$, the set of all ordered pairs with first element from $A$ and second from $B$.

What is a relation from set $A$ to set $B$?

A relation from $A$ to $B$ is any subset of $A \times B$, i.e. a set of ordered pairs $(a,b)$ with $a \in A$ and $b \in B$.

Define the domain and range of a relation.

The domain is the set of all first elements of the ordered pairs; the range is the set of all second elements.

What is a function, and how does it differ from a general relation?

A function $f: A \to B$ is a relation in which every element of the domain $A$ is paired with exactly one element of $B$. Unlike a general relation, no input may have two different outputs.

Define injective (one-to-one), surjective (onto), and bijective functions.

Injective: distinct inputs give distinct outputs ($f(a_1)=f(a_2) \Rightarrow a_1=a_2$). Surjective: every element of the codomain is an image. Bijective: both injective and surjective.

Define a binary operation on a set $S$.

A binary operation $*$ on $S$ is a function $* : S \times S \to S$ that assigns to each ordered pair $(a,b)$ a unique element $a*b \in S$ (closure).

State the four axioms a set $G$ with operation $*$ must satisfy to be a group.

Closure: $a*b \in G$. Associativity: $(a*b)*c = a*(b*c)$. Identity: there exists $e$ with $a*e = e*a = a$. Inverse: each $a$ has $a^{-1}$ with $a*a^{-1} = e$.

What is an Abelian (commutative) group?

A group $(G,*)$ in which the operation is also commutative: $a*b = b*a$ for all $a,b \in G$.

Give the order (dimensions) notation of a matrix and define it.

A matrix with $m$ rows and $n$ columns has order $m \times n$. The element in row $i$, column $j$ is denoted $a_{ij}$.

Define a square matrix, a diagonal matrix, and a scalar matrix.

Square: equal number of rows and columns ($n \times n$). Diagonal: a square matrix with all off-diagonal entries zero. Scalar: a diagonal matrix with all diagonal entries equal.

Define the identity matrix and the null (zero) matrix.

Identity matrix $I$: a diagonal matrix with all diagonal entries $1$; e.g. $I_2 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$. Null matrix: every entry is $0$.

Define symmetric and skew-symmetric matrices.

Symmetric: $A^{t} = A$ (i.e. $a_{ij} = a_{ji}$). Skew-symmetric: $A^{t} = -A$ (i.e. $a_{ij} = -a_{ji}$, so diagonal entries are $0$).

When can two matrices be added, and how is addition performed?

Two matrices can be added only if they have the same order; addition is done entry-wise: $(A+B)_{ij} = a_{ij} + b_{ij}$.

State the condition for the product $AB$ to exist and the order of the result.

If $A$ is $m \times n$ and $B$ is $n \times p$ (columns of $A$ = rows of $B$), then $AB$ exists and has order $m \times p$.

Is matrix multiplication commutative? State the key property.

No. In general $AB \neq BA$; matrix multiplication is associative and distributive over addition but not commutative.

Define the transpose of a matrix and give $(AB)^{t}$.

The transpose $A^{t}$ is formed by interchanging rows and columns. The reversal law holds: $(AB)^{t} = B^{t} A^{t}$.

Give the formula for the determinant of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

$\det A = |A| = ad - bc$.

What is a singular matrix and a non-singular matrix?

A square matrix is singular if $\det A = 0$ (no inverse exists) and non-singular if $\det A \neq 0$ (an inverse exists).

Give the formula for the inverse of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$.

$A^{-1} = \dfrac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, provided $ad - bc \neq 0$.

What are the minor and cofactor of an element $a_{ij}$ of a matrix?

The minor $M_{ij}$ is the determinant obtained by deleting row $i$ and column $j$. The cofactor is $A_{ij} = (-1)^{i+j} M_{ij}$.

What is the adjoint of a matrix, and how is $A^{-1}$ expressed using it?

The adjoint is the transpose of the matrix of cofactors. Then $A^{-1} = \dfrac{1}{\det A}\,\text{adj}(A)$, for $\det A \neq 0$.

Write a system of linear equations in matrix form.

A system can be written as $AX = B$, where $A$ is the coefficient matrix, $X$ the column matrix of variables, and $B$ the column matrix of constants.

Describe the matrix inversion method for solving $AX = B$.

If $A$ is non-singular ($\det A \neq 0$), multiply both sides by $A^{-1}$ to get the unique solution $X = A^{-1}B$.

State Cramer's rule for a system $AX = B$.

For a non-singular system, each unknown is $x_i = \dfrac{\det A_i}{\det A}$, where $A_i$ is $A$ with its $i$-th column replaced by the constants column $B$.

Using Cramer's rule for two equations, write $x$ and $y$ for $\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}$.

$x = \dfrac{\begin{vmatrix} c_1 & b_1 \\ c_2 & b_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}}, \quad y = \dfrac{\begin{vmatrix} a_1 & c_1 \\ a_2 & c_2 \end{vmatrix}}{\begin{vmatrix} a_1 & b_1 \\ a_2 & b_2 \end{vmatrix}}$, with the common denominator $\neq 0$.

What does it mean if $\det A = 0$ when solving a square linear system?

The system is not uniquely solvable; it either has no solution (inconsistent) or infinitely many solutions (dependent equations).

State the quadratic formula for $ax^{2} + bx + c = 0$ ($a \neq 0$).

$x = \dfrac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$.

What is the discriminant of a quadratic equation, and what does it determine?

The discriminant is $\Delta = b^{2} - 4ac$. Its sign determines the nature of the roots of $ax^{2}+bx+c=0$.

Describe the nature of the roots based on the value of the discriminant $b^{2}-4ac$.

If $\Delta > 0$: two distinct real roots ($\Delta$ a perfect square $\Rightarrow$ rational, else irrational). If $\Delta = 0$: real and equal (repeated) roots. If $\Delta < 0$: complex conjugate roots.

When are the roots of $ax^{2}+bx+c=0$ (rational coefficients) rational?

The roots are rational when the discriminant $b^{2}-4ac$ is a perfect square (and $\geq 0$); otherwise they are irrational (or complex).

For roots $\alpha$ and $\beta$ of $ax^{2}+bx+c=0$, give the sum and product of the roots.

Sum: $\alpha + \beta = -\dfrac{b}{a}$. Product: $\alpha \beta = \dfrac{c}{a}$.

How do you form a quadratic equation given its roots $\alpha$ and $\beta$?

$x^{2} - (\alpha + \beta)x + \alpha\beta = 0$, i.e. $x^{2} - (\text{sum of roots})x + (\text{product of roots}) = 0$.

Express $\alpha^{2} + \beta^{2}$ in terms of the sum and product of the roots.

$\alpha^{2} + \beta^{2} = (\alpha + \beta)^{2} - 2\alpha\beta$.

If one root of a real-coefficient quadratic is complex, what is the other root?

Complex roots occur in conjugate pairs; if one root is $a+bi$, the other is its conjugate $a-bi$.