Math 2366: Homework 1 (4 pages) Due as listed on Carmen. Name: .# ....................................................................................................................... Instructions. Write legibly. You may complete this assignment (or any part) on your own paper if you prefer; be sure to label the problems and parts clearly if you do. You may collaborate, but write up your own solutions for submission. Do not use division of labor with collaborators. Submit this homework digitally following the instructions on Carmen. 1. (50 points) Rewrite each statement by (1) choosing an appropriate form from the bulleted list below, and (2) filling in the blanks with a combination of numbers, variables, math symbols, the connective “and”, and the connective “or”. For all If For all , . (universal) , then . (conditional) , if , then . There exist(s) such that There exist(s) such that if For all , there exist(s) There exist(s) (universal conditional) . (existential) , then such that such that for all , . (existential conditional) . (universal existential) . (existential universal) A few examples of possible math symbols are the set membership symbol ∈, the symbol for the set of natural numbers N, the symbol for the set of integers Z, the symbol for the set of rational numbers Q, the symbol for the set of real numbers R, the equality symbol = , the addition symbol + , the multiplication symbol · , the additive inverse symbol − , the divides symbol | , the less-than symbol < , and the less-than-or-equal-to symbol ≤ . Note 1. Some of the statements are true, and some of the statements are false; this is intentional and does not affect how one rewrites the statements. Note 2. Some of the statements may be correctly written using more than one of the allowed forms. 1 Example 1. The square of every integer is greater than or equal to 0. This statement says that a property is true for all elements in a set. So, it begins as a universal: Every integer has the property that its square is greater than or equal to 0. Introduce a universal variable so that we can avoid pronouns when stating the property: For every integer n, n2 is greater than or equal to 0. Reword and group so that we are in one of the allowed forms: For all [integers n], [n2 is greater than or equal to 0]. Replace each group with a symbolic expression (allowing “and” and “or”): For all [n ∈ Z], [n2 ≥ 0]. Remove the groupings to obtain the final answer: For all n ∈ Z, n2 ≥ 0. Example 2. Every real number solution of x 2 − x − 2 = 0 is equal to −1 or 2. This statement says that a property is true for all elements in a set. So, it begins as a universal: Every real number that satisfies the equation x 2 − x − 2 = 0 is equal to −1 or 2. Introduce a universal variable (already implicit in the equation) and introduce a conditional to express the property: For every real number x, if x 2 − x − 2 = 0, then x equals −1 or 2. Reword and group so that we are in one of the allowed forms: For all [real numbers x], if [x 2 − x − 2 = 0], then [x equals −1 or 2]. Replace each group with a symbolic expression (allowing “and” and “or”): For all [x ∈ R], if [x 2 − x − 2 = 0], then [x = −1 or x = 2]. Remove the groupings to obtain the final answer: For all x ∈ R, if x 2 − x − 2 = 0, then x = −1 or x = 2. Alternative solution: If x ∈ R and x 2 − x − 2 = 0, then x = −1 or x = 2. Example 3. There is a maximum rational number. This statement says that a property is true about some particular element of a set. So, it begins as an existential: There exists a rational number such that it is a maximum amongst the rational numbers. Introduce an existential variable to avoid pronouns: There exists a rational number m such that m is a maximum amongst the rational numbers. Saying “m is a maximum amongst the rational numbers” is an assertion about how m compares to other rational numbers: There exists a rational number m such that m is greater than or equal to every rational number. Introduce a universal variable for “every rational number”: There exists a rational number m such that m is greater than or equal to every rational number q. Begin rewording and grouping so that we are in one of the allowed forms: There exists [a rational number m] such that [m is greater than or equal to every rational number q]. We need to reword more because the property asserted is universal: There exists [a rational number m] such that for all [rational numbers q], [m is greater than or equal to q]. Replace each group with a symbolic expression (allowing “and” and “or”): There exists [m ∈ Q] such that for all [q ∈ Q], [q ≤ m]. Remove the groupings to obtain the final answer: There exists m ∈ Q such that for all q ∈ Q, q ≤ m. (Note that we did not write q < m here. This is because q is generic and could be equal to m.) 2 For the items below, you are not required to write out steps like what is shown in the preceding examples. You may write only the final answer. However, it may be helpful to write out steps if you have difficulty with an item. (a) The equation x 2 + x = 1 has a solution in the integers. (b) Every real number equal to its square must be equal to 0 or 1. (c) π 2 is rational provided that π is rational. (d) An integer is negative if its additive inverse is positive. (Notation. The additive inverse of a number x is written −x.) (e) There is a integer n with the property that n divides 51 if n2 + 1 divides 51. (Notation. Writing “m | n” means “m divides n”.) 3 (f) For some real number x, x is negative and the square of x is e. (g) There is a least natural number. (h) Given any real number r , there is an integer z with the properties that z is less than or equal to r and r is less than z + 1. (i) Every real number has a multiplicative inverse. (Note. To say “s is the multiplicative inverse of r ” means “r · s = 1”.) (j) The set of natural numbers has an additive identity. (Note. An additive identity for the set of natural numbers is a natural number m such that the sum of m and any natural number n is equal to n.) 4