FOM BOOK: 1.3

The Introduction Section starts out by explaining proofs, which is basically the root of all of FOM (hehe, math humor). To write a good proof, you must do the following:

  1. Assert a hypothesis and derive a conclusion (If, then statement) 
  2. Interpret the hypothesis based on the problem at hand
  3. Work out the rest of the proof with the interpretation

Get Your Hands Dirty #1

If X is an even integer, and Y is a multiple of X, then Y is even.

First of all, since x is even, it can be rewritten as x = 2n, so Y is a multiple of 2n.

When the problem says that Y is a multiple of X, that means that Y is multiplied by X over and over to create a set of numbers. Say that x=2. Y would then be a multiple of 2, meaning that multiples produced would include 2Y, 4Y, 6Y, 8Y, etc. Multiples produced with just X and Y are XY, 2XY, 3XY, 4XY, etc.

So, since Y is a multiple of 2n (which equals x), Y’s multiples are 2NY, 4NY, 6NY, etc. Because there is a two included in each multiple, each of Y’s multiples in this case are even.

Since Y must be multiplied by 2N in each of it’s multiples, Y will always be even.

Get Your Hands Dirty #2 (Go here to read about conditional statements first!)

A. The sum of two real numbers is always real.

If x and y are both real, then x+y is always real. 

The hypothesis states that x and y must be real, and the conclusion from that is that the sum of x and y is always real.

B. A nonempty set has at least two subsets.

If a set S is nonempty, then S has at least two subsets. 

The hypothesis states that we must first be talking about a nonempty set, and the conclusion from this is that specific nonempty set must then have at least two subsets.

C. Every nonvertical line has a y-intercept.

If a line, y=mx+b, is nonvertical, then y=mx+b must have a y-intercept. 

The hypothesis states the condition that a line must be nonvertical to fit the conclusion that it will always have a y-intercept.

Open Sentencesstatements that are not known to be true or false, as the sentence contains nonspecific variables.

Substitution/Replacement: replacing a variable in an open statement with a specific object.

Truth Value: the characterization of a statement as either true or false

Truth Set: set of values that make an open sentence true

Universe of Discourse: boundaries from which replacement values are chosen (ex: universe = all reals; U = generic universe)

Get Your Hands Dirty #3:

A. 0 < x < 10 (universe = Z

{1,2,3,4,5,6,7,8,9}

B. (t-2)(t^2 – 9) = 0 (universe = R)

{-3, 2, 3)

C. B is a subset of {2,4,6,8,10} with exactly two distinct elements (universe = the set of subsets of {2,4,6,8,10})

For B to be a subset of {2,4,6,8,10}, then B must be either the number 2,4,6,8, or 10. It also must have two distinct elements, which means that each element in the truth set must include two, non repeating numbers mentioned in the set already. The truth set will be really (relatively) large then, including:

{ {2,4}, {2,6}, {2,8}, {2,10}, {4,6}, {4,8}, {4,10}, {6,8},{8,10}, {6,10} }

Relating Subsets and Conditional Sentences: If a conditional statement is true, the truth set for the hypothesis should be a subset of the conclusion. Anything that makes the hypothesis true should then make the conclusion true. Written another way:

\subseteq \!\, B  means “if x \in \!\,A, then x\in \!\,B”

Counterexample:  a value that makes a hypothesis true and a conclusion false. (To read more about what makes conditional statement true or false, go here!)

Even number: an integer that has 2 as a factor; two-directional meaning – all even integers have a factor of two and all numbers with a factor of two are even

IT ALL MAKES SENSE.

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2 thoughts on “FOM BOOK: 1.3

  1. On 1.2C, you claim that a non-vertical line always crosses the y-axis as false. How so? For a linear line (which by definition is infinite) it must eventually cross the y-axis. Moreover, in slope intercept form, y = mx + b, b is the y intercept. The only counterexample I can think of if this were generalized to curves, an asymptotic hyperbola would not cross the y-axis. If you have a counter example of a non-vertical linear equation that doesn’t cross the y-axis, I’d love to know.

    Secondly, on 3C, you correctly state that for B to be a subset of {2,4,6,8,10} but then proceed to list the truth set at {1,2,3,4,5} From my understanding (and I’ve been known to be wrong) it seems the question is asking for two disting element subsets which would look like {2,4} {2,6} {2,8} {2,10} {4,6}…. as the order of a set does not matter {4,2} is repetitive. I’m not exactly sure where 2n came from…

    Happy FOMing!

    • Matt,
      First of all you’re totally right about 2C. For some reason I was thinking about curves and didn’t realize it had to be linear. My bad!

      And 2nd, thanks for clarifying! I really didn’t understand what the question was asking, so I just took a stab at it. That makes sense that two distinct elements would means a set of two elements, not what I was thinking about taking out a two..

      Thanks!
      Kate

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