Question

Let p: a student is in the math club and q: a student is in the debate club. The verbal translation of p ∨ q is: A student is in math club in debate club. p ∨ q is true for a total of students.

Answer 1

The student is asking about logical disjunction in an example involving math and debate clubs. For a similar example with clubs and part-time work, a Venn diagram is used to show the overlapping memberships. In probability, the independence of two events is checked by multiplying their individual probabilities, with a resulting product equal to their joint probability signifying independence.

Step-by-step explanation:

The question is about the logical disjunction operator represented by p ∨ q, which translates to "A student is in the math club or the debate club". The operator ∨ means at least one of the statements is true. When drawing a Venn diagram for a different example involving club membership and part-time work, the diagram would show overlapping circles where one circle represents club membership (C) and the other represents part-time work (PT), with the overlap representing students who do both. For the events A (learning Spanish) and B (learning German), they're considered independent if the probability of A and B occurring together is equal to the product of their individual probabilities, hence P(A AND B) = P(A) ⋅ P(B). To check independence, we would look at the probabilities provided; P(A) = 0.4, P(B) = 0.2, and P(A AND B) = 0.08. Since 0.4 ⋅ 0.2 equals to 0.08, the events A and B are indeed independent.

Answer 2

The correct verbal translation of p ∨ q is 'A student is in the math club or the debate club', representing a logical disjunction. This means the statement is true if the student is in at least one of the clubs. It follows the logical form of a disjunctive syllogism.

Step-by-step explanation:

The verbal translation of p ∨ q is: A student is in the math club or the debate club. It means being a member of at least one of the clubs, not necessarily both. The original statement 'A student is in math club in debate club' seems to have a typo and should use 'or' instead of 'in' to accurately represent p ∨ q.

The symbol '∨' stands for the logical disjunction, which is true if at least one of the propositions p or q is true. If p represents 'a student is in the math club' and q represents 'a student is in the debate club,' then the disjunction p ∨ q is true for any student who is in either or both clubs.

In logical form, the argument provided implies a deductive reasoning structure called disjunctive syllogism. Such an argument form is used when we have two statements joined by 'or', and if one is false, the other must be true. This logical form is crucial in constructing valid deductive inferences.