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I graph the points which satisfy that equation.
Yes.
I understand it this way: graphing an equation (with two variables) means graphing the solutions to the equation on a coordinate system.
I also can't really understand how the solution is obtained to a system of equations.
You do not know how to solve 2 equations in 2 unknowns when they are linear?
I can solve linear systems but I'm feeling lost, meaning I can't understand the process even though I get right answers.
Hi,
Is the variable which ranges over a set called a bound variable?
As for ∈, does it literally mean "a member of" or "ranges over" when generally used with variables? For example I interpret ∈ in { x : x ∈ N} as "ranges over."
I also can't really understand how the solution is obtained to a system of equations. Assume I have two equations with two variables, x and y. Each equation has a solution set, but why is the solution to the system the intersection of equations' solution sets? I know this may sound stupid, but I can't understand it.
Thank you.
I've understood it this way: a variable is a symbol representing an unknown number. It's called a variable since the same symbol can be used to represent other numbers. I recently read that a variable ranges over a given set: does it mean that a variable can stand for any member of that set? If so, how? Is there a solid definition of variable? In basic textbooks a variable is introduced as a symbol for an unknown, but more advanced books seem to assume that you know what a variable is and apply the "range" concept, etc...
Can you tell me what a variable exactly is, in all contexts?
I'm really frustrated, so I thank your assistance.
Hi again,
In the recursive definition of a set, for example:
1) 4∈A
2) If x∈A, then x+1∈A
3) nothing else is in A
Why the step two uses a conditional statement rather than a biconditional one?
I need help with this, thanks...
That sounds like an argument:
1. "MIF forum members" => "really brainy person"
2. there is one member who is not brainy
Therefore, "MIF forum members" does not imply "really brainy person"
Let p="It's raining" and q="It's cloudy" and say p=>q. Semantically, I understand the conditional statement, but I can't understand when one makes a truth table for a conditional. If p is true (i.e. it's raining) and q is false (i.e. it's not cloudy), then the implication is false, so as you said it means that q (being false) does not follow from p.
Now is there a general explanation for unspecified statements p and q: an explanation for why q cannot follow from p when q is false and p is true? If p is false, does that mean, whatever the value of q, that q always follows from p? Is there an alternative word for "follows" since people may also say a conclusion "follows" from a set of premises (as in the latter case the word "follows" refers to logical implication, and in the former case "follows" refers to material implication)?
Thanks for help.
Hi,
What does it mean for a conditional statement to be true or false?
Here's an example, let p="You pass the test" and q="I buy you a car."
p=>q is obviously a conditional statement, but I don't understand the meaning of it being true or false.
A false statement for me is a statement that is not true (or is a lie).
It's not making sense for me to tell that a conditional statement is a lie (or false).
Can someone help me?
Thanks in advance.
Hi,
Just as multiplication is repeated addition, addition is repeated increment.
For example, [3+n = ((n++)++)++], i.e (3+n) is n incremented three times.
Where can I read about this? What keywords to use while searching?
Thanks for help.
Faced with engine problems, Ella Wright made an emergency landing on the beach of the island of Knights and Knaves. The island is inhabited by two distinct groups of people, knights and knaves. Knights always tell the truth and knaves always lie. Ellen decided that her best move was to reach the capital and call for service.
Walking from the beach, she came to an intersection, where she saw two men, A and B, working nearby. After hearing her story, A told Ellen, "The capital is in the mountains, or the road on the right goes to the capital." B then said, "The capital is in the mountains, and the road on the right goes to the capital." Then A looked up and said, "That man is a liar." Shrugging his shoulders, B then said, "If the capital is in the mountains, then the road to the right goes to the capital." Ellen then made a table on the back of her guidebook, thanked the two men, and walked down the road on the left. Did ELlen make the correct decision?
Thank you. So, is it safe to say that a boolean expression is a mathematical statement?
"Expressions that yield the value true or false are boolean expressions."
What's the difference between a statement and a boolean expression?
Thanks for help.
Using my thesaurus,
value: "a numerical quantity measured or assigned or computed"
quantity: "how much there is or how many there are of something that you can quantify"
number: "the property possessed by a sum or total or indefinite quantity of units or individuals"
I don't really understand the definition of a number: Is it safe to say that it's an abstract quantity?
Is it right to think this way: quantity is amount, and a value is a computed or assigned quantity/number.
What is a number then? What's the difference between quantity and number?
Thank you Bob.
Hi,
What's the difference between "value" and "quantity"?
Thanks for help.
Hi,
What about (p⇒q ≡ ¬p∨q)? Why if two statements have the same truth values for every input, then the statements are also semantically the same?
Thanks.
Hi,
I read that two statements are logically equivalent if and only if they have the same truth value for every possible combination.
Does that mean the two statements are also semantically equivalent?
Are (A ⊢ B) and (P ≡ Q) statements/propositions?
Thanks for help.
Hi Bob,
I asked you the first question since I was thinking like this:
Say I have y=x² and A={0, 1, -1, √2, -√2, √3, -√3, 2, -2, etc...}. If x∈R and y∈Z then x∈A and y∈N. The first two constraints imply the other two, but is the relation between R and Z or A and N? I'm feeling somehow confused.
Thanks.
Say I have y=x²-1, where x belongs to the real set, and y to negative integers. Obviously, the graph consists only of one point, which is (0, -1).
Given the restrictions above, is the domain of the relation the set of real numbers or just {0}? Is the codomain equal to the range or the negative integer set?
Are two relations equivalent if they have the same graph but different domains/codomains?
Thanks for help.
Thanks for your response.
Regarding functions, I read that all elements of domain should be associated with elements of codomain, which can have more unassociated elements. Is that the same with relations (other than functions), or can a relation have a domain with unassociated elements?
Hi,
Is this thinking correct: A solution set is the set of all n-tuples (including "1-tuple") which are solutions to an equation or a system of equations. A graph is the solution set of a function.
Thanks.
Hi,
An equation with two variables or more has a solution set which is infinite in size.
A binary relation between set A and set B is a set of ordered pairs. Set A is the domain, and set B is the codomain. The relation is a subset of A×B.
A function is a relation with the property that each element of its domain is related to exactly one element of its codomain.
Are relation, solution set and graph synonyms?
Hi;
The magnitude of a number is its absolute value.
The magnitude of a vector is its length.
The magnitude of a scalar (e.g -10°) is the quantity itself.
Am I thinking right?