Introduction to Mathematical Thinking. Tutorial for Assignment 2

Introduction to Mathematical Thinking.
Tutorial for Assignment 2

1. Simplify the following symbolic statements as you can, leaving your answer in the standard symbolic form. (In case you are not familiar with the notation, I'll answer the first one for you.)
(a) (π > 0) ^ (π<10) [Answer: 0 < π <10]
(b) (p ≥ 7) ^ (p < 12) 7≤p<12
(c) (x>5) ^ (x < 7) 5<x<7
(d) (x < 4) ^ (x <6) x = 4
(e) (y <4) ^ (y2 <9) [−3 < y < 3] y2=9
(f) (x ≥ 0) ^ (x ≤ 0) x = 0
2. Express each of your simplified statements from question 2 in natural English.
3, What strategy would you adopt to show that the conjunction ф1 Ʌ ф2 Ʌ … Ʌ фn is true?
4. What strategy would you adopt to show that the conjunction ф1 Ʌ ф2 Ʌ….

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Ʌ фn is false ?
3. Show all of ф1 Ʌ ф2 Ʌ ….. Ʌ фn are true.
4. Show that are of ф1 Ʌ ф2 Ʌ….Ʌ фn is false.
5, Simplify the following symbolic statements as much as you can, leaving your answer in a standard symbolic form (assuming you are familiar with the notation):
(a) (π >3) V (π > 10) π > 3
(b) (x < 0) V (x > 0) x = 0
(c)(x = 0) V (x > 0) x ≥ 0
(d)(x > 0) V (x ≥ 0) x ≥ 0
(e) (x > 3) v (x2 > 9) (x > 3) v(x < −3) x2 > 9
6. Express each of your simplified statements from question 6 in natural English.
7, What strategy would you adopt to show that the disjunction ф1 Ʌ ф2 Ʌ…. Ʌ фn is true?
8. What strategy would you adopt to show that the disjunction ф1 Ʌ ф2 Ʌ…. Ʌ фn is false ?
7. Show are of ф1 Ʌ ф2 Ʌ….Ʌ фn is true.
8. Show that all of ф1 Ʌ ф2 Ʌ…..Ʌ фn are false.
Let us examine our data. One has been done for you. Two, the simple way to write that is to say 7 less than or equal to p less than 12.
The next one we would write as 5 less than x less than 7. The next one, well if x is less than 4, then it's automatically less than 6. So the second conjunct here is superfluous; we could just write that as x less than 4.
What about the next one? Well, the second conjunct tells us that y is less than 3. But if y satisfies this condition, then it must be less than 4, so the first conjunct is superfluous. The only one that counts is the second one. So we could simply write that as y squared less than 9.
Consider the inequality x≤0. Since x≥0, it follows that 0≤x. However, since x≤0, it follows that 0<x. Thus, there is only one possibility: x=0.
To demonstrate that the conjunction is true, one would show that all of phi 1, phi 2, etc., upto phi n are true.
Fourth, to prove that a conjunction is false is to show that at least one of its components is false. And so, you must find one or more of the components to be false—which would mean that the conjunction itself is also false.
Okay, well that’s that one. If pi is greater than 10, then it’s automatically greater than 3.Therefore in terms of the disjunction, pi greater than 3 dominates—that is, it says more. If we draw a picture, we get the following: 0, 3, 10.
The first disjunct states that x is to the right of this point, and the second disjunct states that x is to the right of that point. The disjunction is true if at least one of these statementsis true. At least one statement will be true if we start at 3.
First of all, only the first disjunct is correct. When we get beyond this point, both disjunctsare correct. So it’s the first one that works: that x is less than 0 or x is greater than 0.
Thus the simplest way to write that is to say that x is not equal to 0. Thus if 0 is negative,or greater than 0, then x is positive. The standard abbreviation for greater than or equal to0 is x ≥ 0.
Once you have x greater than or equal to 0, that's going to dominate the first disjunct interms of a disjunction. So you're going to have x greater than or equal to 0; that makes amore general claim of the two.
And one might make a note that x squared greater than 9 means either x is greater than 3 or x is less than negative 3. So, the statement then becomes simply: x squared greater than 9. That's the one that counts.
As with numbers one and two, when we went through these options we articulated what they were. These options were just to prompt you to think about how you will express them in English.

Okay, well that's that one. In number seven, you have to show that if disjunction is true,then at least one of the disjuncts must be true.
And for the sake of argument, consider the following: If you wish to prove that statement number eight is false, you must demonstrate that all of phi 1 through phi n are false. Okay,well that takes care of numbers seven and eight.
To say that pi is not greater than 3.2 is to say that pi is less than or equal to 3.2.
9. Simplify the following symbolic statements as much as you can, leaving your answer in a standard symbolie from (assuming you are familiar with the notation):
(a) neg(π>3.2) π≤3.2
(b) neg(x < 0) x≥0
(c) neg(x2>0) x=0
(d) neg(x=1) x ne1
(e) neg ψ ψ
10. Express each of your simplified statements from question 9 in natural English.
QUESTION 11
D: "The dollar is strong ", Y: " The Yuan is strong ", T: "New US-China trade agreement signed".
(a) Dollar and Yuan both strong
D Ʌ Y
(b) Yuan weak despite new trade agreement, but Dollar remains strong
neg Y Ʌ T Ʌ D
(c) Dollar and Yuan can't both be strong at same time.
neg (D Ʌ Y)
(d) New trade agreement does not prevent fall in Dollar and Yuan
TɅ negDɅ negY
(e) US-China trade agreement falls but both currencies strong
negT Ʌ D Ʌ Y

When you negate a strict greater than statement, you get less than or equal to. To say that it's not the case that x is negative is to say that x is greater than or equal to 0. These are real numbers because they come from the context of the expression, which is talking about real numbers.
Every real number has a nonnegative square root, with the exception of 0. Thus, the onlyreal number for which it is not the case that x2 ≥ 0 is 0.
The standard symbol for the phrase not x equals 1 is x not equal to 1. When you negate anegation, you get back to the original statement. This way we essentially answered number 10.
We're returning to question 11 now. The answers I received are "Dollar strong and Yuan strong" and "with this word, despite, but, in our mind". Those are just nuanced forms ofconjunction.
We can say that despite the fact that the Yuan is weakening, and there is a trade agreement, the dollar is strong. So despite and but are used to indicate whether something is contrary to expectations or consistent with expectations.
They still say that all three of these things hold together. This one seems fairly straightforward.