Math  /  Algebra

QuestionQ1. (5 points) Consider the open sentence Q(a,b)Q(a, b) There exists a natural number mm such that am2a+3ba m \leq-2 a+3 b.
For each of the following, determine whether it is an open sentence or a statement. If it is an open sentence, indicate the variable(s) it depends on. If it is a statement, give its truth value. No justification is necessary. (a) Q(1,1)Q(1,-1). (b) For all negative integers a,Q(a,b)a, Q(a, b). (c) For all negative integers aa and all positive integers b,Q(a,b)b, Q(a, b).

Studdy Solution
Note that the sentence does not depend on any unspecified variable and thus can be evaluated as either true or false.
To determine its truth value, consider the nature of the inequality and the signs of aa and bb: - For any negative integer aa, 2a-2a is positive. - For any positive integer bb, 3b3b is positive. - The sum 2a+3b-2a + 3b is positive.
Therefore, for any negative integer aa, there exists a natural number m=1m = 1 such that a12a+3ba \cdot 1 \leq -2a + 3b.
Thus, the sentence is a statement and its truth value is true.
Solution Summary: (a) Q(1,1)Q(1,-1) is a statement, and its truth value is false.
(b) "For all negative integers a,Q(a,b)a, Q(a, b)" is an open sentence dependent on bb.
(c) "For all negative integers aa and all positive integers b,Q(a,b)b, Q(a, b)" is a statement, and its truth value is true.

View Full Solution - Free
Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord