Logical Reasoning and Syllogisms Quiz

This is a valid syllogism: No healthy food is fattening (healthy ∩ fattening = ∅). All cakes are fattening (cakes ⊆ fattening). Since cakes are a subset of fattening things, and no fattening thing is healthy, it follows that no cakes are healthy.

Valid syllogism: All animals are mortal (animals ⊆ mortal). All men are animals (men ⊆ animals). By transitivity, if men are a subset of animals and animals are a subset of mortal things, then all men are mortal.

Valid syllogism: No reptiles have fur (reptiles ∩ furry = ∅). All snakes are reptiles (snakes ⊆ reptiles). Since snakes are a subset of reptiles, and no reptile has fur, it follows that no snakes have fur.

Valid syllogism: All kittens are playful (kittens ⊆ playful). Some pets are kittens (pets ∩ kittens ≠ ∅). Since some pets are kittens and all kittens are playful, it follows that some pets are playful.

Valid syllogism: No homework is fun (homework ∩ fun = ∅). Some reading is homework (reading ∩ homework ≠ ∅). Since some reading falls under homework and no homework is fun, it follows that some reading is not fun.

Valid syllogism: All horses have hooves (horses ⊆ hoofed). No humans have hooves (humans ∩ hoofed = ∅). Since all horses have hooves and no humans have hooves, the sets are disjoint, meaning no humans are horses.

Valid syllogism: No lazy students pass exams (lazy ∩ pass = ∅). Some students pass exams (students ∩ pass ≠ ∅). Since some students pass exams and no lazy student passes, the students who pass must be non-lazy, meaning some students are not lazy.

Valid syllogism: All informative things are useful (informative ⊆ useful). Some websites are not useful (websites ∩ useful ≠ websites). Since some websites fall outside the useful set and all informative things are inside it, those websites cannot be informative, meaning some websites are not informative.

From 'All fruit is nutritious' and 'All fruit is tasty', we know that the set of fruit is contained in both nutritious and tasty things. This means there exists an overlap between tasty and nutritious things - specifically, the fruit. Option A is incorrect because we only know that all fruit is both tasty and nutritious, not that ALL tasty things are nutritious. There could be tasty things that are not fruit and not nutritious.

From 'Some mugs are beautiful' and 'All mugs are useful', we can conclude that there exist mugs that are both beautiful and useful (the overlap). These mugs are therefore useful things that are beautiful. Option A is technically true but doesn't represent a valid deduction - it just repeats part of a premise. The correct conclusion is that some useful things are beautiful.

From 'No jug in this cupboard is new' and 'All jugs in this cupboard are cracked', we know that all jugs in this cupboard are both cracked and not new. Therefore, among the cracked items in this cupboard (which include all the jugs), some (specifically the jugs) are not new. This is a valid syllogistic deduction.

From 'All industrious boys have red hair' and 'Some industrious boys are boarders', we know that those industrious boys who are boarders also have red hair. Therefore, some boarders have red hair. Option A is incorrect because we only know about the industrious boys who are boarders, not ALL boarders - there could be boarders who are not industrious and don't have red hair.