This is mostly some basic stuff, but I just want to make sure I am doing these right. I have a hard time understanding what the questions are saying sometimes.(adsbygoogle = window.adsbygoogle || []).push({});

1. The problem statement, all variables and given/known data

1)Explain how we may conclude that if f is a decreasing function, then f(5) < f(3)

Make reference to a logical principle. [Note- By defintion, a function g is decreasing i.f.f the conditional "if x<y, then g(x)>g(y)" is true for any real numbers x and y]

2)Suppose A and B are sets and w is an object, for which it is known that A[tex]\subseteq[/tex]B and w[tex]\in[/tex]A. Explain how we may conclude w[tex]\in[/tex](A[tex]\cap[/tex]B) Make reference to a logical principle.

3)Explain why it is true that if a real number x satisfies |x|>a, but it is not the case that x > a, then x < -a must hold. Make reference to a logical principle.

2. Relevant equations

(p[tex]\wedge[/tex]q) -> pLaw of Simplification

p -> (p[tex]\vee[/tex]q)Law of Addition

[p[tex]\wedge[/tex](p ->q)] -> qModus Ponens

[(p[tex]\vee[/tex]q)[tex]\wedge[/tex] ~q] -> pModus Tollendo Ponens

[(p -> q)[tex]\wedge[/tex]~q] -> ~pModus Tollens

(p -> r) -> [(p[tex]\wedge[/tex]q) -> r]

[~p -> (q[tex]\wedge[/tex]~q] -> ~pLaw of Contradiction

3. The attempt at a solution

1) The definition of a decreasing function states, "if x<y, then g(x)>g(y)" is true for any real numbers x and y. By hypothsis we know f is a decreasing function, so that the preticular case of the definition, "if 5 < 3, then g(5) > g(3)" is known to be true.--That's as far as I can get using an example form the book as a guideline. I'm not sure where to go from here and which logical principle applies to this argument.--

2)--I know how to explain it but I don't know which logical principle to use.--

A[tex]\subseteq[/tex] B includes all the elements in A that are also elements of B. Since it is given w[tex]\in[/tex] A, by this definiton w is also an element of B. A[tex]\cap[/tex]B includes only the elements that A and B share in common. Since w is both an element of A and B, it is one of the elements that A and B share in common, making it an element of A[tex]\cap[/tex]B.

3)If a real number x satisfies |x|>a, that means x < -a or x > a. It is also given that it is not the case that x > a. Therefore by Modus Tollendo Ponens we can conclude that x < -a.

If anyone could help me with finding which logical principles apply to one and two, that'd help a lot.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Basic Propositional Calc

**Physics Forums | Science Articles, Homework Help, Discussion**