xguo9@student.gsu.edu Xuan Guo Lab 5 xguo9@student.gsu.edu Xuan Guo

Contents review some questions of hw2 review power set 24 section 1.4

Question 24 section 1.4 Translate in two ways each of these statements into logical expressions using predicates, quantifiers, and logical connectives. First, let the domain consist of the students in your class and second, let it consist of all people. c) There is a person in your class who cannot swim. d)All students in your class can solve quadratic equations.

Question 24 section 1.4 C(x):”student x is in your class”, S(x):”student x can swim” Q(x):”student x can solve quadratic equations” c) There is a person in your class who cannot swim. d)All students in your class can solve quadratic equations.

Solution C(x):”student x is in your class”, S(x):”student x can swim” Q(x):”student x can solve quadratic equations”

Question 18 section 1.7 Prove that if n is an integer and 3n+2 is even, then n is even using a) a proof by contraposition.

Question 18 section 1.7 Prove that if n is an integer and 3n+2 is even, then n is even using a) a proof by contraposition. Contraposition: if n is odd, then 3n+2 is odd.

Solutions Proof: Contraposition: if n is odd, then 3n+2 is odd. Assume n is odd, then n=2k+1 for some integer k. Then 3n+2=3*(2k+1)+2=6k+5=2*(3k+2)+1 is odd for some integer 3k+2. So if n is odd, then 3n+2 is odd. Then we can get the conclusion if n is an integer and 3n+2 is even, then n is even.

Question 19 section 2.1 19.What is the cardinality of each of these sets? a) {a} b){{a}} c) {a,{a}} d){a,{a},{a,{a}}}

Question 19 section 2.1 19.What is the cardinality of each of these sets? a) {a} 1 b){{a}} c) {a,{a}} 2 d){a,{a},{a,{a}}} 3

Question 23 section 2.1 How many elements does each of these sets have where a and b are distinct elements? a) P({a, b,{a, b}}) b)P({∅,a,{a},{{a}}}) c) P(P(∅))

Question 23 section 2.1 How many elements does each of these sets have where a and b are distinct elements? a) P({a, b,{a, b}}) b)P({∅,a,{a},{{a}}}) c) P(P(∅)) Given a set S, the power set of S is the set of all subsets of the set S. The power set of Sis denoted by P(S).

Question 23 section 2.1 How many elements does each of these sets have where a and b are distinct elements? a) P({a, b,{a, b}}) 8 b)P({∅,a,{a},{{a}}}) 16 c) P(P(∅)) 2

Question 3 section 2.2 3.LetA={1,2,3,4,5} and B={0,3,6}. Find a) A∪B. b)A∩B. c) A−B. d)B−A.

Question 3 section 2.2 3.LetA={1,2,3,4,5} and B={0,3,6}. Find a) A∪B. b)A∩B. b){3} c) A−B. c){1,2,4,5}d){0,6} d)B−A. d){0,6}

Question 13 section 2.2 Prove the second absorption law from Table 1 by showing that if A and B are sets, then A∩(A∪B)=A

Question 13 section 2.2 Prove the second absorption law from Table 1 by showing that if A and B are sets, then A∩(A∪B)=A Hint: A ⊆ A∩(A∪B) and A∩(A∪B) ⊆ A

Question 13 section 2.2 Prove the second absorption law from Table 1 by showing that if A and B are sets, then A∩(A∪B)=A Proof: Suppose x ∈A∩(A∪B). Then x ∈ A and x ∈ A∪B by the definition of intersection. Because x ∈ A, we have proved that the left-hand side is a subset of the right-hand side. Conversely, let x ∈A. Then by the definition of union, x ∈ A∪B as well. Therefore x ∈A∩(A∪B) by the definition of intersection, so the right-hand side is a subset of the left-hand side