Question

Let P: “The figure is a rectangle,” Let q:“The figure is a square”, and r: “The figure is four

sided with four right angles.”
a. Write down the statements p → q and q → r and the converse, inverse and
contrapositive of each one. Determine which are true and which are false.

b. Write down the statement p → r. State the converse. Write out the biconditional
statement. Is this a good definition?
c. True/False (No justification): Since ,p → q and q → r and is true, then by the Law of
Syllogism,p → r is true.
p : q : r :
p → q q → r
p → r
p → q q → r
p → r

Answer 1

a.

(i) p → q: "If the figure is a rectangle, then it is a square." This statement is false because not all rectangles are squares.

Converse: "If the figure is a square, then it is a rectangle." This statement is true because all squares are rectangles.

Inverse: "If the figure is not a rectangle, then it is not a square." This statement is false because a figure can be a square even if it is not a rectangle.

Contrapositive: "If the figure is not a square, then it is not a rectangle." This statement is true because if a figure is not a square, it cannot be a rectangle.

(ii) q → r: "If the figure is a square, then it is four-sided with four right angles." This statement is true because all squares are four-sided with four right angles.

Converse: "If the figure is four-sided with four right angles, then it is a square." This statement is false because a figure can be four-sided with four right angles but not be a square (it could be a rectangle).

Inverse: "If the figure is not a square, then it is not four-sided with four right angles." This statement is false because a figure can not be a square but still be four-sided with four right angles (like a rectangle).

Contrapositive: "If the figure is not four-sided with four right angles, then it is not a square." This statement is true because if a figure is not four-sided with four right angles, it cannot be a square.

b.

p → r: "If the figure is a rectangle, then it is four-sided with four right angles." This statement is true because all rectangles are four-sided with four right angles.

Converse: "If the figure is four-sided with four right angles, then it is a rectangle." This statement is false because a figure can be four-sided with four right angles but not be a rectangle (it could be a square).

The biconditional statement: "The figure is a rectangle if and only if it is four-sided with four right angles." This is not a good definition because a figure can be four-sided with four right angles but not be a rectangle (it could be a square).

c. True. By the Law of Syllogism, if p → q and q → r are true, then p → r is true.

Explanation: