a. If a pentagon has five sides, then it has five vertices.

b. A pentagon has five sides only if it has five vertices.

c. A pentagon has five sides whenever it has five vertices.

d. A triangle has five vertices if it has five sides.

a. p↔q

b. $$\sim$$  p $$\sim$$  q

c. $$\sim$$  q $$\rightarrow$$   $$\sim$$  p

d. q $$\rightarrow$$  p

p $$\leftrightarrow$$  q

q $$\rightarrow$$  p

$$\sim$$  p $$\rightarrow$$   $$\sim$$  q

$$\sim$$  q $$\rightarrow$$   $$\sim$$  p

a. If two lines have the same length, then they are coincident.

A polygon is a rhombus whenever it has four vertices.

Lines meet at a common point if they are concurrent.

Points are coplanar only if they are collinear

a. If x + 3 = 10, then x = 7.

b. A polygon has three sides only if it is a triangle.

c. A figure is a square whenever its four sides are equal.

d. The number 8 is a prime number if its factors are 8 and 1.

a. If 3 x 5 = 15, then 1 + 1 = 0.

b. 1+ 1 = 0 if and only if 3 x 5 = 15.

c. If 3 x 5  15, then 1 + 1   0.

d. If 1 + 1  0, then 3 x 5   15.

true

false

maybe

unknown

a. If air is a gas, then water is a liquid.

b. If water is not a liquid, then air is not a gas.

c. If air is not a gas, then water is not a liquid.

d. Water is a liquid if and only if air is gas.

a. If a triangle has two equal sides, then it is isosceles.

b. If a triangle is not isosceles, then its two sides are not equal.

c. If a triangle is isosceles, then its two sides are not equal.

d. If two sides of a triangle are not equal, then it is not isosceles.

a. Points are not lying on the same plane if and only if they are coplanar.

b. Points are coplanar if and only if they are lying on the same plane.

c. Coplanar are points if and only if they are lying on the same plane.

d. Coplanar are planes if and only if they are points.

true

false

maybe

unknown

inductive

deductive

mathematical intuition

analogy