Discrete Mathmatics :logically equivalent

In summary, the two premises of the sentence are not logically equivalent. The first premise is '(p -> q) or (q -> p)' which can be simplified to 'p or q'. The second premise is '(pVq) or (qVp)' which can be simplified to 'p or q or not p or q'.
  • #1
Ziek_4790
2
0
TL;DR Summary
basically am studying for my exam tomorrow and I got in uni late registration this semester so I can't really get how they solved this question, tried to see the Dr and I couldn't get an appointment or (meet him in general),so please if u can explain this for me I would be grateful, I did understand half of it after spending more than 2 hours on it and I cant figure out rest of the steps.
its just the 2nd question (The equivalence Rule) that I need help with.
1. Consider the statement S =[¬(p ->q)]V[¬(pVq)].
(a) Construct truth tables for S.
(b) Find a simpler expression that is logically equivalent to S.
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  • #2
What part/step is it you're confused about?
 
  • #3
WWGD said:
What part/step is it you're confused about?
this.PNG

exactly these two lines, how did it come to be from (and) to (Or), and where did the p in the left premise came from.
thought that the right premise is (simplification with Idempotent Law ),but then I can't tell how the (and) flipped to (Or).
 
  • #4
It seems they're distributing v over the /\ and then using that pvp==p. (AvB)/\(CvB)=A/\C v. My phone is dying, will get back after charging it.
 
  • #5
One trick is to sub in a + for union/or and a × for intersection ( I also don't know how to \tex them ):
(-p+q)(p+q)=-pp-pq+qp+qq=(-p&p)v(-p&q)v(q&p)v(q&q). The last is the same as q. The 2nd and 3rd term simplify to( changing notation again)

q(p-p)=q&(pv-p)=q&p. But we already have both q, p in the other sentences, which then " absorb" these. Does that make sense?Edit: I will bring someone in a few hours to check, when I am on my PC .
 
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  • #6
A disjunctive statement of the form '(not p) or (not q)' can be exchanged for a logically equivalent denial of a conjunction, as in 'not (p and q)', which is simpler in the sense that it uses the (monary) negation operator once instead of twice. If your symbolic logic system allowed the use of the | operator, which is a binary operator that means 'either not or not', you could simplify by writing (p | q). In computer hardware terms, we would use a NAND gate to replace two inverters and an OR gate.
 
  • #7
Regarding your thread title, please be aware that 'discreet' means not injudiciously overt, while 'discrete' means not continuous. :oldwink:
 
  • #8
sysprog said:
Regarding your thread title, please be aware that 'discreet' means not injudiciously overt, while 'discrete' means not continuous. :oldwink:
Fixed...
 

1. What is discrete mathematics?

Discrete mathematics is a branch of mathematics that deals with discrete objects and structures, rather than continuous ones. It involves the study of mathematical structures such as sets, graphs, and algorithms.

2. What does it mean for two statements to be logically equivalent?

Two statements are said to be logically equivalent if they have the same truth values for all possible combinations of truth values of their component parts. In other words, if they always have the same truth value, regardless of the truth values of their individual parts.

3. How can one prove that two statements are logically equivalent?

There are several methods for proving logical equivalence, including truth tables, logical equivalences, and logical deductions. These methods involve analyzing the truth values of the component parts of the statements and showing that they are always equivalent.

4. What are some examples of logically equivalent statements?

Some examples of logically equivalent statements include "if and only if" statements, such as "A if and only if B," "not both" statements, such as "either A or B, but not both," and "contrapositive" statements, such as "if not A, then not B."

5. Why is understanding logical equivalence important in discrete mathematics?

Understanding logical equivalence is important in discrete mathematics because it allows us to simplify complex statements and proofs. By identifying logically equivalent statements, we can reduce them to simpler forms and make them easier to analyze and understand.

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