Question

1. Given two conditions P and Q, there are four ways in which they might be related to each other.

P is necessary but not sufficient for Q.

P is sufficient but not necessary for Q.

P is both necessary and sufficient for Q.

P is neither necessary nor sufficient for Q.

For each problem below, do the following;

i. Determine whether P is necessary, sufficient, both necessary and sufficient, or neither necessary nor sufficient for Q.

ii. Justify your answer for i. above.

iii. Write the associated implication, if any.

a) P = “each side is equal in length to each of the others”, Q = “being a square”

b) P = “having four sides”, Q = “being a triangle”

c) P = “being a parallelogram”, Q = “having four sides”

d) P = “the number can be represented as 2n+1 for some integer n”, Q = “the number is odd”

e) P = “the number is even”, Q = “the number is divisible by 8”

f) P = “having more than one laptop”, Q = “having three laptops”

g) P = “being a younger sibling”, Q = “having an older sister”

h) P = “being a younger brother”, Q = “having an older sibling”

Answer 1

a) P = “each side is equal in length to each of the others”, Q = “being a square”
P is necessary but not sufficient for Q,Because rhombus has all of its side equal.

b) P = “having four sides”, Q = “being a triangle”
P is neither necessary nor sufficient for Q.Because triangle has only three sides.

c) P = “being a parallelogram”, Q = “having four sides”
P is sufficient but not necessary for Q.Because it can be any 4 sided structure not necessarily a parallelogram.

d) P = “the number can be represented as 2n+1 for some integer n”, Q = “the number is odd”
P is both necessary and sufficient for Q.
Because every number of form 2n+1 is an odd number.

e) P = “the number is even”, Q = “the number is divisible by 8”
P is necessary but not sufficient for Q.Because all numbers which is divisible by 8 are even but 4 is not divisible by 8.

f) P = “having more than one laptop”, Q = “having three laptops”
P is necessary but not sufficient for Q.Because having two laptops is not having three laptops.

g) P = “being a younger sibling”, Q = “having an older sister”
P is necessary but not sufficient for Q.There may have an older brother.

h) P = “being a younger brother”, Q = “having an older sibling”
P is sufficient but not necessary for Q.