Prove that the inverse and converse of conditional statements are logically equivalent.
Show that conditional statement is logically equivalent to its contrapositive.
Define tautology, contradiction and contingency with example.
Construct the truth table for the proposition
Find the negation of the following:
All students study for exams.
Some politicians are honest and sincere.
Every McDonald’s serves French fries.
Some people attend college.
There are people who believe in UFO’s.
Check whether the statement “Good foods are not expensive” and “Cheap foods are not good” are logically equivalent or not.
Discuss the direct proof, indirect proof, and proof by contradiction with suitable example.
Define trivial proof and vacuous proof. Using direct proof, show that for every positive integer n, n³+ n is even.
Prove that 3 divides n³+2n whenever n is a non-negative integer.
Define universal and existential quantifier. Using direct proof, show that the difference of two rational numbers is rational.
Prove that the product xy is odd if and only if both x and y are odd integers.
Give a direct proof that if m and n are both perfect squares, then nm is also a perfect square.
Express the statement “Every student in the class has studied computer” using quantifier.
Test the validity of the given argument.
Smoking is healthy. If smoking is healthy then cigarettes are prescribed by physicians. Therefore, cigarettes are prescribed by physicians.
Construct an argument using rules of inference to show that hypotheses “If it does not rain or if it is not foggy then the sailing race will be held and the lifesaving demonstration will go on”. “If sailing race is held then trophy will be awarded” and “The trophy was not awarded” imply the conclusion “It rained”.
Prove or disprove the validity of argument: Every living thing is a plant or an animal. Hari’s dog is alive and it is not a plant. All animals have heart. Hence, Hari’s dog has heart.
There is someone in this class who has visited Pokhara. Everyone who has been in Pokhara visits Fewa Lake. Therefore, someone in this class has visited Fewa Lake.
If x is a parent of y, then x is older than y. If x is mother of y, then x is parent of y. Lulu is mother of Fifi. Therefore, Lulu is older than Fifi.
If a teacher teaches DM or DSA, then he is considered to be a TCS teacher. If he is a TCS teacher, then he teaches GT. He does not teach GT. Therefore, he does not teach DSA.
Use mathematical induction to prove that 1.1! + 2.2! +………n.n! = (n+1)! – 1, where n is a positive integer.
