the choices are valid or unvalid Decide whether or not each of the following con

Question

the choices are valid or unvalid

Decide whether or not each of the following conclusions, together with the given premises, constitute a valid argument Note: There are 5 parts (a-e.) to this problem. Select only one response for each part. Premise 1. "Discrete structures is fun or not many students like Discrete structures." Premise 2. "If Computer Science is easy, then Discrete structures is not fun." a. Discrete structures is not fun or Computer Science is not easy. Choose... > b. If not many students like Discrete structures, then either Computer Science is not easy or Discrete structures is not fun. Choose... C. Not many students like Discrete structures, if Computer Science is not easy. Choose... d. Computer Science is not easy, if many students like Discrete structures. Choose.. e, Computer Science is not easy or Discrete structures is fun. Choose...

Explanation / Answer

Question: Given
Premise 1. "Discrete sturctures is fun or not many students like Discrete Structures"
Premise 2."If Computer Science is easy,then Discrete Structures is not fun."
Solution: From the given premises Let’s make a few definitions:

• We will use ds to represent “Discrete sturctures” and cs to represent “Computer Science”.
• Given n {ds, cs}, F (n) will mean that the discipline n is fun.
• Given n {ds, cs}, E (n) will mean that the discipline n is easy.
• Given n {ds, cs}, S (n) will mean that many students enjoy discipline n.
As one may expect, ¬S (n) will mean that few students enjoy n.

Now, we can symbolically express the two assumptions for premises:
(i) (F (ds) ¬S (ds))

(ii) (E (cs) ¬F (ds))

Observe that (by the definition of implication and by contraposition,
respectively) these two are equivalent to the following:
(i) (S (ds) F (ds))
(ii) (F (ds) ¬E (cs))

We are now ready to choose the following claims:
(a) “Discrete Structures is not Fun or Computer science is not easy.”
Symbolically,we express this claim as (¬F (ds) ¬E (cs)).
If we convert this to an implication, we get (F (ds) ¬E (cs)),
which is one of our hypotheses. The claim is therefore valid.

(b) “If not many students like Discrete Structures, then either Computer Science is not
easy or Discrete Structures is not fun.”
Here we have a complex claim:
(¬S (ds) (¬E (cs) ¬F (ds))).
If we convert the right side into an implication,we get

(¬S (ds) (F (ds) ¬E (cs))).

However,since (F(ds) ¬E (cs)) is one of our hypotheses, this claim becomes (¬S (ds) T),

which is equivalent to (S (l) T), which (by the domination law) is always true.
Hence, the claim is valid.

(c) “computer science is not easy if many students like Discrete Structures.”
This can be written as (S (ds) ¬E (cs)).
Since we know from above that (S (ds) F (ds)), and that (F (ds) ¬E (cs)),
we conclude that this claim is valid.

(d) Computer Science is not easy ,if not many students like Discrete Structures.
This claim can be written as (¬E (cs) ¬S (ds)).
By contraposition,we obtain (S (ds) E (cs)).
Again, we know from our hypotheses that
(S (ds) F (ds)), and that (F (ds) ¬E (cs)),
so we conclude that this claim is invalid.

(e) “Computer Science is not easy or Discrete Stuctures is fun.”
We can write this claim as (¬E (cs) F (ds)),
which is equivalent (by the definition of implication) to (E (cs) F (ds)).
However, this claim contradicts (E (cs) ¬F (ds)), which is one of our hypotheses.
Therefore the claim is invalid.

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