"Some computer science majors take discrete math". S is the domain of all college students C(x) means "x is a computer science major" D(x) means "x takes discrete math". Can someone please explain why the following statement is wrong according to the TA? There exists x in S such that C(x) implies D(x)

Faith Welch 2022-07-15 Answered
"Some computer science majors take discrete math"
S is the domain of all college students C(x) means "x is a computer science major" D(x) means "x takes discrete math"
Can someone please explain why the following statement is wrong according to the TA?
There exists x in S such that C(x) implies D(x)
I don't understand
EDIT: I can see how it could be: There exists x in S such that C(x) AND D(x), but I don't see why an implication is wrong for SOME x in S
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Answers (2)

grocbyntza
Answered 2022-07-16 Author has 25 answers
Step 1
The statement is wrong, among other reasons, because the statement
C(x) implies D(x) is true for any x that is not a computer science student, and for any x that takes discrete math. In particular, the statement
There exists x S such that C(x) implies D(x). (1)
would be true if nobody is a computer science major. However, I think most people agree that
Some computer science majors take discrete math should be considered false if there are no computer science majors at all.
Step 2
Also, the statement (1) would be true if there is at least one person taking discrete math, whether or not that person is a computer science major. So, in a university in which at least one person takes Discrete Math, but no computer science major does, the statement "There exists x∈S such that C(x) implies D(x)" would be true, but the statement "Some computer science majors take discrete math" would be false.
What you need to remember is that an implication is true if the antecedent is false, or if the consequent is true. You don't need the antecedent to be true.
What you actually want is:
There exists x S such that C(x) and D(x).

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skilpadw3
Answered 2022-07-17 Author has 4 answers
Step 1
Remember that C ( x ) D ( x ) is logically equivalent to ¬ C ( x ) D ( x ), which in words is ‘x is not a computer science major, or x takes discrete math (or both)’. Your statement, therefore, can be translated into English as
there is a college student who takes discrete math or is not a computer science major (or both).
Step 2
Suppose that there is exactly one college student who takes discrete math, and he’s not a computer science major. Then your statement is true, but ‘some computer science majors take discrete math’ isn’t. Thus, yours can’t be the same statement.

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