3. (12 points) Express each of these statements using quantifiers and the given
ID: 3595658 • Letter: 3
Question
3. (12 points) Express each of these statements using quantifiers and the given predicates, along with arithmetic operations (e.g. x2, 2x). Then form the negation of the statement, so that no negation operator is to the left of a quantifier. Next, express this negation you formed in simple and precise English. (a) Statement: For all real numbers a, if 21, then >0. Allowed predicates: G(a, b)-"a > b", Eq(a, b)-"a Allowed predicate: Eq(a, b) = "a = b". Reminder: the reciprocal of a real number a is a real number b such that ab (b) Statement: For all integers a, b, c, if a - b is even and b - c is even, then a - cis even. (c) There is a real number with no reciprocal. 1 Allowed predicates: Eq(a, b)= "a=b" (d) There is a real number x, such that for all real numbers y, 2x+y = 5. Allowed predicates: Eq(a, b) = "a = b".Explanation / Answer
Answer for Question 1:
Given Statement x, if x^2 >=1 , then x > 0
Eq(a,b) = "a = b"
G(a,b) = "a > b"
Consider a = 1 and b = 1
G(1,1) = 1 > 1 condition fails
Eq(1,1) => 1 = 1 condition true
Consider a = 2 and b = 1
G(2,1) => 2 > 1 condition true
Eq(2,1) => 2 = 1 condition fails
Answer for Question 2:
Given Statement a - b is even, b - c is even, and a- c is even
Eq(a,b) = "a = b"
Consider a = 3 , b = 5 , c = 3
a - b = 3 - 5 = -2 even
b - c = 5 - 3 = 2 even
a - c = 3 - 3 = 0 even
Based above consideration given statement is true.
Answer for Question 3:
Given Statement The reciprocal of a real number a is a
real number b such that ab = 1
Eq(a,b) = "a = b"
a reciprocal means 1/a
consider a = 1, b = 1 so ab = 1, 1 * 1/1 =1 condition is true
condider a = 2, b = 1 so ab = 1, 2 * 1 / 2 = 1 condition is fails because a and b are difrenet.
Answer for Question 5:
Given Statement 2x + y = 5;
Eq(a,b) = > a = b
Consider a = 2 , b = 1
Eq(2,1) = 2 * 2 + 1 = 5 true allowed predicate fails.
Consider a = 3, b = -1
Eq(3,-1) = 2 * 3 - 1 = 5 true allowed predicate true.
Related Questions
drjack9650@gmail.com
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.