discreate mathmatics

Question

Express the following English statement using quantifiers and find out the negation of the quantified

expression. Convert the negation of the quantified expression into English statement.

“All the student in the sophomore have attended the lecture by Freeman Dyson”.

Answer 1

Since you're asked to use quantifiers, I'll assume predicates are on the table as well.

Sx = is a student

Px = is a sophomore

Dx = attended the lecture by Freeman Dyson

so we want, "for all x, if x is a student and a sophomore, they attended the lecture by Freeman Dyson."

Using V for the universal quantifier, this is Vx((Sx & Px) --> Dx). The negation of this is ~Vx((Sx & Px) --> Dx), which, in English, is "it's not the case that all the sophomore students attended the lecture by Freeman Dyson," which is equivalent to "at least one sophomore student didn't attend the lecture," or, more naturally, "not all the sophomore students attended the lecture by..."

Hope that helps!