# 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.
 ﻿A B﻿ ﻿ ﻿T ﻿T ﻿F ﻿T F﻿ T﻿ ﻿F T﻿ T﻿
 ﻿F F F﻿
Find a logical expression, using only that is equivalent to XOR. Also, write an expression using XOR that is equivalent to .