Exam 1 Exercises

Problem 1)

The statement is false if and only if a) is false b) is true c) is false d) is true

Problem 2)

For the sentence write its a) inverse b) converse c) contrapositive Which of these are equivalent?

Problem 3)

Define the word cardinality as the number of elements in a set. What is the cardinality of a) {10, 9, 8, 7, 6} b) The set of all positive whole numbers less than 20 and divisible by 5.

Problem 4)

Draw the truth-table for and the table for . Are these equivalent?

Problem 5)

Given and what is the reason which justifies the statement .

Problem 6)

Given that
  • If Adam score a goal then Bill and Carly will play defense.
  • If Carly and Dolores play defense Adam will not score another goal.
  • Adam scored a goal and then another goal.
Prove that Dolores did not play defense. Write a two-column proof where A1 represents "Adam scores a goal", A2 represents "Adam scores a second goal", B represents "Bill plays defense", C represents "Carly plays defense", and D represents "Dolores plays defense".

Problem 7)

Define the very important logical operator, XOR, which we pronounce "ex-or". It is short for "exclusive or". It is used often in Computer Science and Cryptography applications, and we use the symbol to represent it. It has the following truth-table.
Find a logical expression, using only that is equivalent to XOR. Also, write an expression using XOR that is equivalent to .