# (Homework Solution): (10*15 + 2*15 + 2*15 + 4* 15 # 275 Points) 3.1 In the questions below write the negation

(10*15 + 2*15 + 2*15 + 4* 15 # 275 Points) 3.1 In the questions below write the negation of the statement. (Don’t write “It is not true that. 3.1.1 It is Thursday and it is cold. .1.2 Iwill go to the play or read a book, but not both. 3.1.3 If it is rainy, then we go to the movies. 3.1.4 Some bananas are yellow. 1.5 All integers ending in the digit 7 are odd. 3.1.6 No tests are easy. .1.7 Roses are red and violets are blue 3.1.8 Some skiers do not speak Swedish. 3.1.9 It is Thursday and it is cold. .1.10 Iwill go to the play or read a book, but not both. 3.2 Explain why the negation of 3.2.1 “Al and Bill are absent” is not “Al and Bill are present” 3.2.2 “Some students in my class use e-mail” is not “Some students in my class do not use e-mail. 3.3 On the island of knights and knaves you encounter two people, A and B. Person A says, “B is a knave.” Person B says: 3.3.1 We are both knights.” Determine whether each person is a knight or a knave. 3.3.2 At least one of us is a knight.” Determine whether each person is a knight or a knave. 3.4 Using cfor “it is cold”, r for “it is rainy”, and wfor “it is windy”, write in symbolic form 3.4.1 t is neither cold nor dry 3.4.2 “t is rainy if it is not cold 3.4.3 “To be windy it is necessary that it be cold” 3.4.4 “t is rainy only if it is windy and cold” in symbols

3.1.1 It is not Thursday, or it is not cold.

3.1.2 Either I will got to the play and read a book, or I will not go to the play and not read a book.

3.1.3 If it is rainy, we don’t go to the movies.

3.1.4 None of the bananas are yellow

3.1.5 At least one integer ending in 7 is not odd.

3.1.6 At least one test is easy.

3.1.7 Roses aren’t red, or violets aren’t blue.

3.1.8 All of the skiers speak Swedish.

3.1.9 It is not Thursday, or it is not cold.

3.1.10 Either I will got to the play and read a book, or I will not go to the play and not read a book.

3.2.1 “Al and Bill are absent” implies that both Al and Bill are absent. This statement is negated even if one of them (or both) is present (not absent). So a valid negation of the statement may be “At least one out of Al and Bill is present”. The statement “Al and Bill are present” only mentions the specific case when both are present.

3.2.2 “Some students in my class use e-mail” implies that there is at least one student in my class who uses e-mail. The statement is negated by saying that “No student in my class uses e-mail”. The negation proposed in the question may even be valid when the statement is true. Consider when 50% students use e-mail. In this case, the given statement and the provided negation both are true, which implies that the given negation is not a valid negation of the given statement.

3.3.1 A is a Knight, and B is a Knave. If A is a Knight and he says that “B is a Knave”, then B must be a Knave. And then if B says that “We are both Knights”, then he must be lying. And when A is a Knight and B is A Knave, then B’s statement that “We are both Knights” is false indeed.

3.3.2 A is a Knave, and B is a Knight. If A is a Knave then his statement that “B is a Knave” must be false, i.e., B must be a Knight. And then B must be telling the truth. And indeed B’s statement “At least one of us is a Knight” checks up.

3.4.1  . Using d for “it is dry”.

3.4.2

3.4.3

3.4.4

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