Question

1. indicate whether the arguments below are valid or invalid. support your answer by drawing a diagram.

(a)
all discrete mathematics students can tell a valid argument from an invalid one.
all thoughtful people can tell a valid argument from an invalid one.
∴ all discrete mathematics students are thoughtful.

(b)
no vegetarians eat meat.
all vegans are vegetarian.
∴ no vegans eat meat.

(c)
all math 216 students are geniuses.
all geniuses have hidden super powers.
∴ all math 216 students have hidden super powers.

Explanation:

Part (a)

The premises state two groups (discrete math students, thoughtful people) share a trait, but this does not mean one group is a subset of the other. A Euler diagram would show two separate subsets inside the set of "people who can tell valid/invalid arguments," with no overlap required. This makes the conclusion unsupported.

Part (b)

The premises establish that vegans are a subset of vegetarians, and vegetarians are entirely outside the set of "meat eaters." A Euler diagram would show the vegan set entirely inside the vegetarian set, which is outside the meat-eater set, so the conclusion logically follows.

Part (c)

The premises state Math 216 students are a subset of geniuses, and geniuses are a subset of "people with hidden super powers." A Euler diagram would nest Math 216 students inside geniuses, which is nested inside the super power set, so the conclusion is logically necessary.

Answer:

(a) Invalid
(b) Valid
(c) Valid