Discrete Structure · Unit 1
Fundamentals of Logic and Propositions
Exam-focused notes for Fundamentals of Logic and Propositions (Discrete Structure, BIT152): what the TU syllabus asks and how it has actually been tested, with 6 solved past questions from this unit.
What this unit covers
- Propositions and non-propositions
- Logical connectives and truth tables
- Tautology and contradiction
- Converse, inverse, and contrapositive
- Logical equivalence
- Quantifiers and predicate logic
- Negation of statements
Quantifiers and predicate logic
Why do we need quantifiers? List any three rules of inferences. Prove that $2\sqrt{2}$ is irrational using proof by contradiction. [2+3+5]
In propositional logic, we can only deal with specific propositions (true or false statements). However, many mathematical and logical statements involve variables and express properties over a range of objects. Quantifiers allow us to express such statemen...
Full solved answer →Define proposition. Convert the following sentences to predicate: (a) Some kind hearted peoples do still exist. (b) Student who study hard and do the homework get good marks in exam. [5]
A proposition is a declarative statement that is either true or false, but not both simultaneously. It has a definite truth value (T or F). Examples: - "2 + 2 = 4" (True proposition) - "The sky is green" (False proposition) - "Close the door!" (NOT a propos...
Full solved answer →Propositions and non-propositions
Give any four examples that are not propositions. Assume the premises, all over smart persons are stupid, children of stupid persons are naughty, John is over smart, Sam is children of John. Using rules of inferences, show that Sam is naughty.[10]
A proposition is a declarative statement that is either true or false, but not both. The following are not propositions because they are questions, commands, exclamations, or paradoxes with no definite truth value: Example Reason -------------------- 1 "Wha...
Full solved answer →Tautology and contradiction
What is tautology? Show $(p \land q) \rightarrow (p \lor q)$ is a tautology. [5]
A tautology is a propositional formula (compound statement) that is always true regardless of the truth values assigned to its component variables. In other words, every row of its truth table yields True (T). --- We need to evaluate (p ∧ q) → (p ∨ q) for a...
Full solved answer →Negation of statements
List the negations of following statements: (a) He has passed the exam. (b) All dogs are loyal. (c) Some medicine has side effect. (d) If you study then you will pass the exam. (e) Open the door. [5]
Negation: He has not passed the exam. --- Negation: Some dogs are not loyal. Note: The negation of a universal statement ("All A are B") is an existential statement ("Some A are not B").) --- Negation: No medicine has side effect. (i.e., All medicines have ...
Full solved answer →Converse, inverse, and contrapositive
What do you mean by converse, inverse, and contrapositive? Show that the sentences "if it is hot today then today is Sunday" and "if it is not Sunday then today is not hot" are logically equivalent. [5]
Let the original conditional statement be: P → Q (If P, then Q) Name Form Description ------------------------- Converse Q → P Swap the hypothesis and conclusion Inverse ¬P → ¬Q Negate both hypothesis and conclusion Contrapositive ¬Q → ¬P Swap AND negate bo...
Full solved answer →Make Unit 1 stick
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