Theory question 115
LogicConsider the following statements:
X: All lazy students are careless.
Y: Most dull students are lazy.
(a) Illustrate the information above on a Venn diagram.
(b) Using the Venn diagram, or otherwise, determine whether or not each of the following statements is valid. (i) Stella is careless ⇒ Stella is lazy. (ii) Osei is lazy ⇒ Osei is careless. (iii) Kojo is dull ⇒ Kojo is lazy. (iv) Kwame is lazy ⇒ Kwame is dull. (v) Adwoa is lazy and dull ⇒ Adwoa is careless. (vi) Fiifi is careless and dull ⇒ Fiifi is lazy.
(c) If x is a positive integer, determine the least value of x for which 13 + 2x = 3 (mod 8).
Show the model answer
(a) U = students. Set L (lazy) drawn completely inside set C (careless). Set D (dull) overlaps L, with most of D inside L and part of D outside L.
(b)(i) Not valid (a careless student need not be lazy).
(ii) Valid (L ⊂ C).
(iii) Not valid (only most dull students are lazy).
(iv) Not valid (some lazy students are not dull).
(v) Valid (lazy ⇒ careless).
(vi) Not valid (careless and dull does not force lazy).
(c) 2x ≡ 3 − 13 = −10 ≡ 6 (mod 8). x = 3 works: 13 + 6 = 19 = 2 × 8 + 3. Least x = 3.
'All A are B' means draw A inside B; 'most' means an overlap, not a subset.