Discrete Mathmatics :logically equivalent

[Ziek_4790]

TL;DR

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.

 

[WWGD]

What part/step is it you're confused about?

 

[Ziek_4790]

What part/step is it you're confused about?

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).

 

[WWGD]

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.

 

[WWGD]

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 .

 

[sysprog]

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.

 

[sysprog]

Regarding your thread title, please be aware that 'discreet' means not injudiciously overt, while 'discrete' means not continuous.

 

[Mark44]

Regarding your thread title, please be aware that 'discreet' means not injudiciously overt, while 'discrete' means not continuous.

Fixed...