# 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.

_{1}represents "Adam scores a goal", A_{2}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 .

A | B | |

T | T | F |

T | F | T |

F | T | T |

F | F | F |

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