Mathematics for Computing - Logical Operators and Boolean Algebra Lab Notes

Faculty of Computing - IT1130 Mathematics for Computing

• Course Details: Year $1$, Semester $1$ ($2026$)
• Document Context: Lab Sheet $03$ focusing on Logic and Base Conversions

Relational Expressions and Logical Operators

• Logical Operators Definition: The truth value of a Boolean expression is determined by combining the truth values of its component expressions using logical operators.
  • OR ($||$): This operator returns $true$ if at least one component expression is $true$.
  • AND (&&): This operator returns $true$ only if all component expressions are $true$.
  • NOT ($\sim$): This operator inverts the truth value; $true$ becomes $false$, and $false$ becomes $true$.

Section 1.1 - Basic Evaluation

• Evaluation of relational assignments:
  • $a = 10 > 3$: The condition is $true$, so $a = 1$.
  • $b = 5 == 7$: The condition is $false$, so $b = 0$.
  • $c = 8 ~= 8$: The condition "not equal to" is $false$, so $c = 0$.

Section 1.2 - Logical Operations

• Given Variables:

  • $a = 4 < 6$ (Result: $true$, value is $1$)
  • $b = 10 >= 12$ (Result: $false$, value is $0$)

• Evaluation Tasks:

  • a \text{ && } b: Evaluates to 1 \text{ && } 0, resulting in $0$ ($false$).
  • $a \text{ || } b$: Evaluates to $1 \text{ || } 0$, resulting in $1$ ($true$).
  • $\sim a$: Evaluates to $\sim 1$, resulting in $0$ ($false$).
  • \sim (a \text{ && } b): Evaluates to \sim (1 \text{ && } 0) = \sim (0), resulting in $1$ ($true$).

Operator Precedence Hierarchy

• Order of Operations: Unless brackets are used to modify the order, logical operators are evaluated in the following sequence:

  1. NOT ($\sim$): Evaluated first.
  2. AND (&&): Evaluated second.
  3. OR ($||$): Evaluated last.

• Evaluation Example 1:

  • Given: $a = 1, b = 0, c = 1$
  • Expression: a \text{ || } b \text{ && } c
  • Step 1 (AND): b \text{ && } c = 0 \text{ && } 1 = 0
  • Step 2 (OR): $a \text{ || } 0 = 1 \text{ || } 0 = 1$
  • Result: $1$

• Evaluation Example 2:

  • Given: $a = 1, b = 0, c = 1$
  • Expression: (a \text{ || } b) \text{ && } c
  • Step 1 (Brackets): $a \text{ || } b = 1 \text{ || } 0 = 1$
  • Step 2 (AND): 1 \text{ && } c = 1 \text{ && } 1 = 1
  • Result: $1$

Section 2.1 - Number Systems and Pattern Recognition

• Task: Evaluate the function $dec2bin(2^n - 1)$ for specific values of $n$.
• Evaluations:
  • For $n = 3$: $2^3 - 1 = 7$. Output: $111$
  • For $n = 4$: $2^4 - 1 = 15$. Output: $1111$
  • For $n = 5$: $2^5 - 1 = 31$. Output: $11111$
  • For $n = 6$: $2^6 - 1 = 63$. Output: $111111$
• Pattern Identification: The binary output consists of a string of ones where the count of ones is exactly equal to the value of $n$.

Section 3.1 - Basic Boolean Operations

• Evaluation Q2:
  • 1 \text{ && } 0: Output is $0$.
  • $1 \text{ || } 0$: Output is $1$.
  • $\sim 1$: Output is $0$.

Section 3.2 - Boolean Expression Evaluation

• Variables: $A = 1$, $B = 0$, $C = 1$
• Evaluations Q3:
  • (A \text{ && } B) \text{ || } C
    • Calculation: (1 \text{ && } 0) \text{ || } 1 = 0 \text{ || } 1 = 1
  • \sim (A \text{ || } B) \text{ && } C
    • Calculation: \sim (1 \text{ || } 0) \text{ && } 1 = \sim (1) \text{ && } 1 = 0 \text{ && } 1 = 0

Section 3.3 - Verifying Boolean Identities

• Identity Check Q4: Evaluate if (A \text{ && } (A \text{ || } B)) == A
  • Trial with $A = 1, B = 0$:
    • (1 \text{ && } (1 \text{ || } 0)) = (1 \text{ && } 1) = 1.
    • Since the result is $1$, and $A = 1$, the equation holds true ($1 == 1$).
  • Note: It is encouraged to test other combinations of values for $A$ and $B$ to verify this Absorption Law in Boolean Algebra.

Section 4.1 - Simple Decision Making Applications

• Scenario Q5: Modeling a real-life eligibility check.
• Given Data:
  • $age = 20$
  • $hasID = 1$ (representing $true$)
• Logic Model: (age >= 18) \text{ && } hasID
• Step-by-step Evaluation:
  • $(20 >= 18)$ evaluates to $true$ ($1$).
  • 1 \text{ && } 1 evaluates to $1$.
• Interpretation: The output $1$ ($true$) indicates that the condition for the decision (e.g., entry or legal eligibility) has been met.

Section 4.2 - Multiple Conditions Applications

• Scenario Q6: Determining qualification based on academic performance and attendance.
• Given Data:
  • $marks = 75$
  • $attendance = 80$
• Logic Model: (marks >= 50) \text{ && } (attendance >= 75)
• Step-by-step Evaluation:
  • $(75 >= 50)$ is $true$ ($1$).
  • $(80 >= 75)$ is $true$ ($1$).
  • 1 \text{ && } 1 results in $1$.
• Testing Variability: Students should understand how altering values (e.g., setting $marks = 40$ or $attendance = 60$) changes the outcome, demonstrating how multiple criteria are enforced simultaneously by the AND operator.