Discrete Mathematics Answers

USING MATHEMATICAL INDUCTION P(n)=1+2²+2ⁿ+........+2ⁿ= 2ⁿ⁺¹ - 1

Are these system specifications consistent? ”The file system is not locked only if new messages will be queued. If the file system is not locked, then the system is functioning normally, and conversely. For new messages are not queued, it is necessary that they will be sent to the message buffer. Whenever the file system is not locked, new messages will be sent to the message buffer. New messages will not be sent to the message buffer. 

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 An orientation of a graph G = (V, E) is any directed graph G0 = (V, E0 ) arising by replacing each edge {u, v} ∈ E by the directed edge (u, v) or by the directed edge (v, u). Show that for every planar graph there is an orientation such that each vertex has at most five outgoing edges.

If we have two graph G = (V, E) and G0 = (V, E0 ) on the same set V of vertices we call the union G ∪ G0 the graph (V, E ∪ E0 ). Next we have two claim about the chromatic number χ(G ∪ G0 ) of G ∪ G0 .Prove that χ(G∪G0 ) ≤ χ(G)∗χ(G0 ). Hint: consider assigning a color to each vertex in V based on its color in colorings of G and G0 .

Indian Cricket team was doing catching practice .The fielding coach informed then that they will throw ball to 4 other players done this ,but others did not throw any ball to catch.how many players received the ball including first player,if no one passes more than once,and if the practice ends after there have been 16 players who catches the bAll but did not throw to anyone

He relation (a,b) such that "a" and "b" have the same age ,is defined on the sets of all people.Is it equivalent relation?

Check whether the relation defined by{(a,b)}| a is cousin of b}defined on the sets of all human being is an equivalence or not?

translate the statement "The product of any two positive integers is always positive"into logical expression(using Nested Quantifiers)

Using propositional-formula and logical connectives, convert these sentences into symbolic form. 

1. No doctors are enthusiastic;  

You are enthusiastic.  

Therefore, you are not a doctor  

1. Dictionaries are useful;  

Useful books are valuable.  

Therefore, dictionaries are valuable.   

1. Some healthy people are fat;  

No unhealthy people are strong.  

Therefore, some fat people are not strong.  

1. All, who are anxious to learn, work hard;  

Some of these boys work hard.  

Therefore, some of these boys are anxious to learn.   

1. All lions are fierce;  

Some lions do not drink coffee.  

Therefore, some creatures that drink coffee are not fierce.  

1.  No misers are generous;  

Some old men are ungenerous.  

 Therefore, some old men are misers.  

1. All young lambs jump;  

No young animals are healthy, unless they jump. 

Therefore, all young lambs are healthy.  

1. No professors are ignorant.  

All ignorant people are vain.  

Therefore, no professors are vain.   

1. All bees are unfriendly;  

No butterflies are unfriendly.  

Therefore, butterflies are not bees.