Comprehensive Study Notes • 2 Modules
Date: January 16, 2026
A declarative statement that is either True or False, but never both. It represents a fact. Examples: 'It rained yesterday', 'Islamabad is the capital of Pakistan'. Non-examples: Questions, commands, or open sentences like x^2 = 13.
A compound statement that is True only when both components are True. Otherwise, it is False.
A compound statement that is False only if both inputs are False. If at least one input is True, the result is True.
Represents the opposite truth value of the original statement. If P is True, ~P is False, and vice versa.
'Not AND'. The result is False only if both inputs are True. For all other combinations, the result is True.
'Not OR'. The result is True only if both inputs are False. For all other cases, the result is False.
The result is True if the inputs are different (one True, one False). The result is False if the inputs are the same.
False only when the hypothesis (p) is True and the conclusion (q) is False. In all other cases, the result is True.
True only when both p and q have the same truth value (both True or both False). If they differ, the result is False.
A compound proposition that is always True, regardless of the truth values of its individual variables.
A compound proposition that is always False, regardless of the truth values of its individual components.
A compound proposition that is neither a tautology nor a contradiction; its truth value depends on variable inputs.
~q → ~p
Negate both conclusion and hypothesis, and swap positions. Logically equivalent to the original conditional.
q → p
Swap the positions of the hypothesis and the conclusion.
~p → ~q
Negate both the hypothesis and the conclusion without changing positions.
p ∧ q
~p ∧ ~q (or ~(p ∨ q))
Date: N/A
For any integer a (dividend) and positive integer d (divisor), there exist unique integers q (quotient) and r (remainder) such that a = dq + r, where 0 ≤ r < d.
A system containing integers from 0 to m-1.
Integers a and b are congruent modulo m if m divides (a - b).
The remainder (r) cannot be negative. It must always be between 0 and d-1.
When dividing a negative number (e.g., -11 / 3), calculate q such that the remainder becomes positive. Example: -11 = 3(-4) + 1, so q=-4, r=1.