PU-4 BСА-2
BCA (IV- SEMESTER)
EXAMINATION, JUNE-2024
MATHEMATICS-II
Time: 3 Hours Full Marks: 70
Note: The questions are equal value.
Answer any five questions.
Group-A
Q1. Explain the concepts of tautologies, contradictions, and logical equivalence in propositional logic. Provide examples and truth tables for each concept.
Q2. Construct truth tables for the propositions P^(QQ)P\land (Q\lor\ negQ) P^(Q^-Q) and (PV-P)\neg (P\lor\neg P)-(PV-P). Show which are tautologies, contradictions, and logically equivalent statements.
Q3. Define logical connectives and their role in forming compound statements. Create a truth table for the compound statement (PQ)^(QP) (P\right arrow Q) \land (neg Q \rightarrow \ neg P) (P -> Q) ^( neg Q neg P) and analyze its logical equivalence.
Q4. Describe an axiom system for propositional calculus. How do truth tables help in determining logical validity?
Q5. Use an axiom system to prove the logical equivalence of (P vee Q) leftrightarrow( neg P Q) (PlorQ) leftright arrow (neg P rightarrow Q) and and verify using truth tables. (P vee Q) leftrightarrow( neg P -> Q )
Group-B
Q6. Explain the concepts of universal and existential quantifiers in predicate calculus. Provide an example of each type of quantifier and discuss their application in formal reasoning.
Q7. Express and interpret the statements forall x \in R * (x * 2 >= 0) ^ 1 for all x\in mathbb (R) ( x ^ 2 \qes 0) forall x \in R(x * 2 >= 0) and exists x \in R(x * 2 = - 1) exists x\in \ mathbb (R) (x^ * 2 = - 1 ) exists x \in R(x * 2 = - 1) using universal and existential quantifiers.
Q8. Discuss partial order relations, least upper bounds (lub), and greatest lower bounds (glb). Provide an example of a partially ordered set and determine its lub and glb.
Q9. Define and discuss the basic properties of semigroups, groups, rings, integral domains, and fields. Provide examples for each structure and explain their key properties.
Q10. Given the vectors vI = (1, 2) ^ nu mathbf{v) 0.1 = (1, 2) v * 1 = (1, 2) and v2= (3, 4) | mathbf \ v\ \ 2 = (3, 4) v2=( 3 , 4), determine if wey are linearty independent. Find the basis for the vector space spanned by these vectors.
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