A college student makes the following statement:If I receive an A in both Calculus I and

Which of the following sentences are statements? For those that are, indicate the truth value. (a) The integer 123 is prime. (b) The integer 0 is even. (c) Is 5 2 = 10? (d) x 2 4 = 0. (e) Multiply 5x + 2 by 3. (f) 5x + 3 is an odd integer. (g) What an impossible question!

Consider the sets A, B,C and D below. Which of the following statements are true? Give an explanation for each false statement. A = {1, 4, 7, 10, 13, 16,...} C = {x Z : x is prime and x = 2} B = {x Z : x is odd} D = {1, 2, 3, 5, 8, 13, 21, 34, 55,...} (a) 25 A (b) 33 D (c) 22 / A D (d) C B (e) B D (f) 53 / C.

Which of the following statements are true? Give an explanation for each false statement. (a) (b) {} (c) {1, 3}={3, 1} (d) = {} (e) {} (f) 1 {1}.

Consider the open sentence P(x) : x(x 1) = 6 over the domain R. (a) For what values of x is P(x) a true statement? (b) For what values of x is P(x) a false statement?

For the open sentence P(x):3x 2 > 4 over the domain Z, determine: (a) the values of x for which P(x) is true. (b) the values of x for which P(x) is false.

For the open sentence P(A) : A {1, 2, 3} over the domain S = P({1, 2, 4}), determine: (a) all A S for which P(A) is true. (b) all A S for which P(A) is false. (c) all A S for which A {1, 2, 3}=.

Let P(n): n and n + 2 are primes. be an open sentence over the domain N. Find six positive integers n for which P(n) is true. If n N such that P(n) is true, then the two integers n, n + 2 are called twin primes. It has been conjectured that there are infinitely many twin primes.

Let P(n) : n2+5n+6 2 is even. (a) Find a set S1 of three integers such that P(n) is an open sentence over the domain S1 and P(n) is true for each n S1. (b) Find a set S2 of three integers such that P(n) is an open sentence over the domain S2 and P(n) is false for each n S2.

Find an open sentence P(n) over the domain S = {3, 5, 7, 9} such that P(n) is true for half of the integers in S and false for the other half.

Find two open sentences P(n) and Q(n), both over the domain S = {2, 4, 6, 8}, such that P(2) and Q(2) are both true, P(4) and Q(4) are both false, P(6) is true and Q(6) is false, while P(8) is false and Q(8) is true.

State the negation of each of the following statements. (a) 2 is a rational number. (b) 0 is not a negative integer. (c) 111 is a prime number.

Complete the truth table in Figure 2.16.

State the negation of each of the following statements. (a) The real number r is at most 2. (b) The absolute value of the real number a is less than 3. (c) Two angles of the triangle are 45o. (d) The area of the circle is at least 9. (e) Two sides of the triangle have the same length. (f) The point P in the plane lies outside of the circle C.

State the negation of each of the following statements. (a) At least two of my library books are overdue. (b) One of my two friends misplaced his homework assignment. (c) No one expected that to happen. (d) Its not often that my instructor teaches that course. (e) Its surprising that two students received the same exam score.

Complete the truth table in Figure 2.17.

For the sets A = {1, 2, , 10} and B = {2, 4, 6, 9, 12, 25}, consider the statements P: A B. Q: |A B| = 6. Determine which of the following statements are true. (a) P Q (b) P ( Q) (c) P Q (d) ( P) Q (e) ( P) ( Q).

Let P: 15 is odd. and Q : 21 is prime. State each of the following in words, and determine whether they are true or false. (a) P Q (b) P Q (c) ( P) Q (d) P ( Q).

Let S = {1, 2,..., 6} and let P(A) : A {2, 4, 6}=. and Q(A) : A = . be open sentences over the domain P(S). (a) Determine all A P(S) for which P(A) Q(A) is true. (b) Determine all A P(S) for which P(A) (Q(A)) is true. (c) Determine all A P(S) for which (P(A)) (Q(A)) is true.

Consider the statements P: 17 is even. and Q: 19 is prime. Write each of the following statements in words and indicate whether it is true or false. (a) P (b) P Q (c) P Q (d) P Q.

For statements P and Q, construct a truth table for (P Q) ( P)

Consider the statements P : 2 is rational. and Q : 22/7 is rational. Write each of the following statements in words and indicate whether it is true or false. (a) P Q (b) Q P (c) ( P) ( Q) (d) ( Q) ( P).

Consider the statements: P: 2 is rational. Q: 2 3 is rational. R: 3 is rational. Write each of the following statements in words and indicate whether the statement is true or false. (a) (P Q) R (b) (P Q) ( R) (c) (( P) Q) R (d) (P Q) ( R).

Suppose that {S1, S2} is a partition of a set S and x S. Which of the following are true? (a) If we know that x / S1, then x must belong to S2. (b) Its possible that x / S1 and x / S2. (c) Either x / S1 or x / S2. (d) Either x S1 or x S2. (e) Its possible that x S1 and x S2.

Two sets A and B are nonempty disjoint subsets of a set S. If x S, then which of the following are true? (a) Its possible that x A B. (b) If x is an element of A, then x cant be an element of B. (c) If x is not an element of A, then x must be an element of B. (d) Its possible that x / A and x / B. (e) For each nonempty set C, either x A C or x B C. (f) For some nonempty set C, both x A C and x B C.

A college student makes the following statement: If I receive an A in both Calculus I and Discrete Mathematics this semester, then Ill take either Calculus II or Computer Programming this summer. For each of the following, determine whether this statement is true or false. (a) The student doesnt get an A in Calculus I but decides to take Calculus II this summer anyway. (b) The student gets an A in both Calculus I and Discrete Mathematics but decides not to take any class this summer. (c) The student does not get an A in Calculus I and decides not to take Calculus II but takes Computer Programming this summer. (d) The student gets an A in both Calculus I and Discrete Mathematics and decides to take both Calculus II and Computer Programming this summer. (e) The student gets an A in neither Calculus I nor Discrete Mathematics and takes neither Calculus II nor Computer Programming this summer.

A college student makes the following statement: If I dont see my advisor today, then Ill see her tomorrow. For each of the following, determine whether this statement is true or false. (a) The student doesnt see his advisor either day. (b) The student sees his advisor both days. (c) The student sees his advisor on one of the two days. (d) The student doesnt see his advisor today and waits until next week to see her.

The instructor of a computer science class announces to her class that there will be a well-known speaker on campus later that day. Four students in the class are Alice, Ben, Cindy and Don. Ben says that hell attend the lecture if Alice does. Cindy says that shell attend the talk if Ben does. Don says that he will go to the lecture if Cindy does. That afternoon exactly two of the four students attend the talk. Which two students went to the lecture?

Consider the statement (implication): If Bill takes Sam to the concert, then Sam will take Bill to dinner. Which of the following implies that this statement is true? (a) Sam takes Bill to dinner only if Bill takes Sam to the concert. (b) Either Bill doesnt take Sam to the concert or Sam takes Bill to dinner. (c) Bill takes Sam to the concert. (d) Bill takes Sam to the concert and Sam takes Bill to dinner. (e) Bill takes Sam to the concert and Sam doesnt take Bill to dinner. (f) The concert is canceled. (g) Sam doesnt attend the concert.

Let P and Q be statements. Which of the following implies that P Q is false? (a) ( P) ( Q) is false. (b) ( P) Q is true. (c) ( P) ( Q) is true. (d) Q P is true. (e) P Q is false.

Consider the open sentences P(n):5n + 3 is prime. and Q(n):7n + 1 is prime., both over the domain N. (a) State P(n) Q(n) in words. (b) State P(2) Q(2) in words. Is this statement true or false? (c) State P(6) Q(6) in words. Is this statement true or false?

In each of the following, two open sentences P(x) and Q(x) over a domain S are given. Determine the truth value of P(x) Q(x) for each x S. (a) P(x) : |x| = 4; Q(x) : x = 4; S = {4, 3, 1, 4, 5}. (b) P(x) : x 2 = 16; Q(x) : |x| = 4; S = {6, 4, 0, 3, 4, 8}. (c) P(x) : x > 3; Q(x):4x 1 > 12; S = {0, 2, 3, 4, 6}.

In each of the following, two open sentences P(x) and Q(x) over a domain S are given. Determine all x S for which P(x) Q(x) is a true statement. (a) P(x) : x 3 = 4; Q(x) : x 8; S = R. (b) P(x) : x 2 1; Q(x) : x 1; S = R. (c) P(x) : x 2 1; Q(x) : x 1; S = N. (d) P(x) : x [1, 2]; Q(x) : x 2 2; S = [1, 1]

In each of the following, two open sentences P(x, y) and Q(x, y) are given, where the domain of both x and y is Z. Determine the truth value of P(x, y) Q(x, y) for the given values of x and y. (a) P(x, y): x 2 y2 = 0. and Q(x, y): x = y. (x, y) {(1, 1), (3, 4), (5, 5)}. (b) P(x, y): |x|=|y|. and Q(x, y): x = y. (x, y) {(1, 2), (2, 2), (6, 6)}. (c) P(x, y): x 2 + y2 = 1. and Q(x, y): x + y = 1. (x, y) {(1, 1), (3, 4), (0, 1), (1, 0)}.

Each of the following describes an implication. Write the implication in the form if, then. (a) Any point on the straight line with equation 2y + x 3 = 0 whose x-coordinate is an integer also has an integer for its y-coordinate. (b) The square of every odd integer is odd. (c) Let n Z. Whenever 3n + 7 is even, n is odd. (d) The derivative of the function f (x) = cos x is f  (x) = sin x. (e) Let C be a circle of circumference 4. Then the area of C is also 4. (f) The integer n3 is even only if n is even.

Let P : 18 is odd. and Q : 25 is even. State P Q in words. Is P Q true or false?

Let P(x) : x is odd. and Q(x) : x 2 is odd. be open sentences over the domain Z. State P(x) Q(x) in two ways: (1) using if and only if and (2) using necessary and sufficient.

For the open sentences P(x) : |x 3| < 1. and Q(x) : x (2, 4). over the domain R, state the biconditional P(x) Q(x) in two different ways.

Consider the open sentences: P(x) : x = 2. and Q(x) : x 2 = 4. over the domain S = {2, 0, 2}. State each of the following in words and determine all values of x S for which the resulting statements are true. (a) P(x) (b) P(x) Q(x) (c) P(x) Q(x) (d) P(x) Q(x) (e) Q(x) P(x) (f) P(x) Q(x).

For the following open sentences P(x) and Q(x) over a domain S, determine all values of x S for which the biconditional P(x) Q(x) is true. (a) P(x) : |x| = 4; Q(x) : x = 4; S = {4, 3, 1, 4, 5}. (b) P(x) : x 3; Q(x):4x 1 > 12; S = {0, 2, 3, 4, 6}. (c) P(x) : x 2 = 16; Q(x) : x 2 4x = 0; S = {6, 4, 0, 3, 4, 8}.

In each of the following, two open sentences P(x, y) and Q(x, y) are given, where the domain of both x and y is Z. Determine the truth value of P(x, y) Q(x, y) for the given values of x and y. (a) P(x, y) : x 2 y2 = 0 and; Q(x, y) : x = y. (x, y) {(1, 1), (3, 4), (5, 5)}. (b) P(x, y) : |x|=|y| and; Q(x, y) : x = y. (x, y) {(1, 2), (2, 2), (6, 6)}. (c) P(x, y) : x 2 + y2 = 1 and; Q(x, y) : x + y = 1. (x, y) {(1, 1), (3, 4), (0, 1), (1, 0)}.

Determine all values of n in the domain S = {1, 2, 3} for which the following is a true statement: A necessary and sufficient condition for n3+n 2 to be even is that n2+n 2 is odd.

Determine all values of n in the domain S = {2, 3, 4} for which the following is a true statement: The integer n(n1) 2 is odd if and only if n(n+1) 2 is even.

Let S = {1, 2, 3}. Consider the following open sentences over the domain S: P(n): (n+4)(n+5) 2 is odd. Q(n): 2n2 + 3n2 + 6n2 > (2.5)n1. Determine three distinct elements a, b, c in S such that P(a) Q(a) is false, Q(b) P(b) is false, and P(c) Q(c) is true.

Let S = {1, 2, 3, 4}. Consider the following open sentences over the domain S: P(n): n(n1) 2 is even. Q(n): 2n2 (2)n2 is even. R(n): 5n1 + 2n is prime. Determine four distinct elements a, b, c, d in S such that (i) P(a) Q(a) is false; (ii) Q(b) P(b) is true; (iii) P(c) R(c) is true; (iv) Q(d) R(d) is false.

Let P(n): 2n 1 is a prime. and Q(n): n is a prime. be open sentences over the domain S = {2, 3, 4, 5, 6, 11}. Determine all values of n S for which P(n) Q(n) is a true statement.

For statements P and Q, show that P (P Q) is a tautology.

For statements P and Q, show that (P ( Q)) (P Q) is a contradiction.

For statements P and Q, show that (P (P Q)) Q is a tautology. Then state (P (P Q)) Q in words. (This is an important logical argument form, called modus ponens.)

For statements P, Q and R, show that ((P Q) (Q R)) (P R) is a tautology. Then state this compound statement in words. (This is another important logical argument form, called syllogism.)

Let R and S be compound statements involving the same component statements. If R is a tautology and S is a contradiction, then what can be said of the following? (a) R S (b) R S (c) R S (d) S R.

For statements P and Q, the implication (P) (Q) is called the inverse of the implication P Q. (a) Use a truth table to show that these statements are not logically equivalent. (b) Find another implication that is logically equivalent to ( P) ( Q) and verify your answer.

Let P and Q be statements. (a) Is (P Q) logically equivalent to (P) (Q)? Explain. (b) What can you say about the biconditional (P Q) ((P) (Q))?

For statements P, Q and R, use a truth table to show that each of the following pairs of statements is logically equivalent. (a) (P Q) P and P Q. (b) P (Q R) and (Q) ((P) R).

For statements P and Q, show that (Q) (P (P)) and Q are logically equivalent.

For statements P, Q and R, show that (P Q) R and (P R) (Q R) are logically equivalent.

Two compound statements S and T are composed of the same component statements P, Q and R. If S and T are not logically equivalent, then what can we conclude from this?

Five compound statements S1, S2, S3, S4 and S5 are all composed of the same component statements P and Q and whose truth tables have identical first and fourth rows. Show that at least two of these five statements are logically equivalent.

Verify the following laws stated in Theorem 2.18: (a) Let P, Q and R be statements. Then P (Q R) and (P Q) (P R) are logically equivalent. (b) Let P and Q be statements. Then (P Q) and (P) (Q) are logically equivalent.

Write negations of the following open sentences: (a) Either x = 0 or y = 0. (b) The integers a and b are both even.

Consider the implication: If x and y are even, then x y is even. (a) State the implication using only if. (b) State the converse of the implication. (c) State the implication as a disjunction (see Theorem 2.17). (d) State the negation of the implication as a conjunction (see Theorem 2.21(a)).

For a real number x, let P(x) : x 2 = 2. and Q(x) : x = 2. State the negation of the biconditional P Q in words (see Theorem 2.21(b)).

Let P and Q be statements. Show that [(P Q) (P Q)] (P Q).

Let n Z. For which implication is its negation the following? The integer 3n + 4 is odd and 5n 6 is even.

For which biconditional is its negation the following? n3 and 7n + 2 are odd or n3 and 7n + 2 are even.

Let S denote the set of odd integers and let P(x) : x 2 + 1 is even. and Q(x) : x 2 is even. be open sentences over the domain S. State x S, P(x) and x S, Q(x) in words.

Define an open sentence R(x) over some domain S and then state x S, R(x) and x S, R(x) in words.

State the negations of the following quantified statements, where all sets are subsets of some universal set U: (a) For every set A, A A = . (b) There exists a set A such that A A.

State the negations of the following quantified statements: (a) For every rational number r, the number 1/r is rational. (b) There exists a rational number r such that r 2 = 2.

Let P(n): (5n 6)/3 is an integer. be an open sentence over the domain Z. Determine, with explanations, whether the following statements are true: (a) n Z, P(n). (b) n Z, P(n).

Determine the truth value of each of the following statements. (a) x R, x 2 x = 0. (b) n N, n + 1 2. (c) x R, x 2 = x. (d) x Q, 3x 2 27 = 0. (e) x R, y R, x + y + 3 = 8. (f) x, y R, x + y + 3 = 8. (g) x, y R, x 2 + y2 = 9. (h) x R, y R, x 2 + y2 = 9

The statement For every integer m, either m 1 or m2 4. can be expressed using a quantifier as: m Z, m 1 or m2 4. Do this for the following two statements. (a) There exist integers a and b such that both ab < 0 and a + b > 0. (b) For all real numbers x and y, x = y implies that x 2 + y2 > 0. (c) Express in words the negations of the statements in (a) and (b). (d) Using quantifiers, express in symbols the negations of the statements in both (a) and (b).

Let P(x) and Q(x) be open sentences where the domain of the variable x is S. Which of the following implies that ( P(x)) Q(x) is false for some x S? (a) P(x) Q(x) is false for all x S. (b) P(x) is true for all x S. (c) Q(x) is true for all x S. (d) P(x) Q(x) is false for some x S. (e) P(x) ( Q(x)) is false for all x S.

Let P(x) and Q(x) be open sentences where the domain of the variable x is T . Which of the following implies that P(x) Q(x) is true for all x T ? (a) P(x) Q(x) is false for all x T . (b) Q(x) is true for all x T . (c) P(x) is false for all x T . (d) P(x) ( (Q(x)) is true for some x T . (e) P(x) is true for all x T . (f) ( P(x)) ( Q(x)) is false for all x T .

Consider the open sentence P(x, y,z): (x 1)2 + (y 2)2 + (z 2)2 > 0. where the domain of each of the variables x, y and z is R. (a) Express the quantified statement x R, y R, z R, P(x, y,z) in words. (b) Is the quantified statement in (a) true or false? Explain. (c) Express the negation of the quantified statement in (a) in symbols. (d) Express the negation of the quantified statement in (a) in words. (e) Is the negation of the quantified statement in (a) true or false? Explain.

Consider quantified statement For every s S and t S, st 2 is prime. where the domain of the variables s and t is S = {3, 5, 11}. (a) Express this quantified statement in symbols. (b) Is the quantified statement in (a) true or false? Explain. (c) Express the negation of the quantified statement in (a) in symbols. (d) Express the negation of the quantified statement in (a) in words. (e) Is the negation of the quantified statement in (a) true or false? Explain.

Let A be the set of circles in the plane with center (0, 0) and let B be the set of circles in the plane with center (1, 1). Furthermore, let P(C1,C2): C1 and C2 have exactly two points in common. be an open sentence where the domain of C1 is A and the domain of C2 is B. (a) Express the following quantified statement in words: C1 A, C2 B, P(C1,C2). (2.30) (b) Express the negation of the quantified statement in (2.30) in symbols. (c) Express the negation of the quantified statement in (2.30) in words.

For a triangle T , let r(T ) denote the ratio of the length of the longest side of T to the length of the smallest side of T . Let A denote the set of all triangles and let P(T1, T2): r(T2) r(T1). be an open sentence where the domain of both T1 and T2 is A. (a) Express the following quantified statement in words: T1 A, T2 A, P(T1, T2). (2.31) (b) Express the negation of the quantified statement in (2.31) in symbols. (c) Express the negation of the quantified statement in (2.31) in words.

Consider the open sentence P(a, b): a/b < 1. where the domain of a is A = {2, 3, 5} and the domain of b is B = {2, 4, 6}. (a) State the quantified statement a A, b B, P(a, b) in words. (b) Show the quantified statement in (a) is true.

Consider the open sentence Q(a, b): a b < 0. where the domain of a is A = {3, 5, 8} and the domain of b is B = {3, 6, 10}. (a) State the quantified statement b B, a A, Q(a, b) in words. (b) Show the quantified statement in (a) is true.

Give a definition of each of the following and then state a characterization of each. (a) Two lines in the plane are perpendicular. (b) A rational number.

Define an integer n to be odd if n is not even. State a characterization of odd integers.

Define a triangle to be isosceles if it has two equal sides. Which of the following statements are characterizations of isosceles triangles? If a statement is not a characterization of isosceles triangles, then explain why. (a) If a triangle is equilateral, then it is isosceles. (b) A triangle T is isosceles if and only if T has two equal sides. (c) If a triangle has two equal sides, then it is isosceles. (d) A triangle T is isosceles if and only if T is equilateral. (e) If a triangle has two equal angles, then it is isosceles. (f) A triangle T is isosceles if and only if T has two equal angles.

By definition, a right triangle is a triangle one of whose angles is a right angle. Also, two angles in a triangle are complementary if the sum of their degrees is 90. Which of the following statements are characterizations of a right triangle? If a statement is not a characterization of a right triangle, then explain why. (a) A triangle is a right triangle if and only if two of its sides are perpendicular. (b) A triangle is a right triangle if and only if it has two complementary angles. (c) A triangle is a right triangle if and only if its area is half of the product of the lengths of some pair of its sides. (d) A triangle is a right triangle if and only if the square of the length of its longest side equals to the sum of the squares of the lengths of the two smallest sides. (e) A triangle is a right triangle if and only if twice of the area of the triangle equals the area of some rectangle.

Two distinct lines in the plane are defined to be parallel if they dont intersect. Which of the following is a characterization of parallel lines? (a) Two distinct lines 1 and 2 are parallel if and only if any line 3 that is perpendicular to 1 is also perpendicular to 2. (b) Two distinct lines 1 and 2 are parallel if and only if any line distinct from 1 and 2 that doesnt intersect 1 also doesnt intersect 2. (c) Two distinct lines 1 and 2 are parallel if and only if whenever a line intersects 1 in an acute angle , then also intersects 2 in an acute angle . (d) Two distinct lines 1 and 2 are parallel if and only if whenever a point P is not on 1, the point P is not on 2.

Construct a truth table for P (Q ( P)).

Given that the implication (Q R) ( P) is false and Q is false, determine the truth values of R and P.

Find a compound statement involving the component statements P and Q that has the truth table given in Figure 2.18.

Determine the truth value of each of the following quantified statements: (a) x R, x 3 + 2 = 0. (b) n N, 2 3 n. (c) x R, |x| = x. (d) x Q, x 4 4 = 0. (e) x, y R, x + y = . (f) x, y R, x + y = x 2 + y2.

Rewrite each of the implications below using (1) only if and (2) sufficient. (a) If a function f is differentiable, then f is continuous. (b) If x = 5, then x 2 = 25.

Let P(n): n2 n + 5 is a prime. be an open sentence over a domain S. (a) Determine the truth values of the quantified statements n S, P(n) and n S, P(n) for S = {1, 2, 3, 4}. (b) Determine the truth values of the quantified statements n S, P(n) and n S, P(n) for S = {1, 2, 3, 4, 5}. (c) How are the statements in (a) and (b) related?

(a) For statements P, Q and R, show that ((P Q) R) ((P (R)) (Q)). (b) For statements P, Q and R, show that ((P Q) R) ((Q ( R)) (P)).

For a fixed integer n, use Exercise 2.91 to restate the following implication in two different ways: If n is a prime and n > 2, then n is odd.

For fixed integers m and n, use Exercise 2.91 to restate the following implication in two different ways: If m is even and n is odd, then m + n is odd.

For a real-valued function f and a real number x, use Exercise 2.91 to restate the following implication in two different ways: If f  (x) = 3x 2 2x and f (0) = 4, then f (x) = x 3 x 2 + 4.

For the set S = {1, 2, 3}, give an example of three open sentences P(n), Q(n) and R(n), each over the domain S, such that (1) each of P(n), Q(n) and R(n) is a true statement for exactly two elements of S, (2) all of the implications P(1) Q(1), Q(2) R(2) and R(3) P(3) are true, and (3) the converse of each implication in (2) is false.

Do there exist a set S of cardinality 2 and a set {P(n), Q(n), R(n)} of three open sentences over the domain S such that (1) the implications P(a) Q(a), Q(b) R(b) and R(c) P(c) are true, where a, b, c S and (2) the converses of the implications in (1) are false? Necessarily, at least two of these elements a, b and c of S are equal.

Let A = {1, 2,..., 6} and B = {1, 2,..., 7}. For x A, let P(x):7x + 4 is odd. For y B, let Q(y):5y + 9 is odd. Let S = {(P(x), Q(y)) : x A, y B, P(x) Q(y) is false}. What is |S|?

Let P(x, y,z) be an open sentence, where the domains of x, y and z are A, B and C, respectively. (a) State the quantified statement x A, y B, z C, P(x, y,z) in words. (b) State the quantified statement x A, y B, z C, P(x, y,z) in words for P(x, y,z) : x = yz. (c) Determine whether the quantified statement in (b) is true when A = {4, 8}, B = {2, 4} and C = {1, 2, 4}.

Let P(x, y,z) be an open sentence, where the domains of x, y and z are A, B and C, respectively. (a) Express the negation of x A, y B, z C, P(x, y,z) in symbols. (b) Express (x A, y B, z C, P(x, y,z)) in words. (c) Determine whether (x A, y B, z C, P(x, y,z)) is true when P(x, y,z) : x + z = y. for A = {1, 3}, B = {3, 5, 7} and C = {0, 2, 4, 6}.

Write each of the following using if, then. (a) A sufficient condition for a triangle to be isosceles is that it has two equal angles. (b) Let C be a circle of diameter 2/. Then the area of C is 1/2. (c) The 4th power of every odd integer is odd. (d) Suppose that the slope of a line is 2. Then the equation of is y = 2x + b for some real number b. (e) Whenever a and b are nonzero rational numbers, a/b is a nonzero rational number. (f) For every three integers, there exist two of them whose sum is even. (g) A triangle is a right triangle if the sum of two of its angles is 90o. (h) The number 3 is irrational.