p: Every square is a rectangle.
q: Every rhombus is a kite.
The truth values of $p\to q\ and\ p\leftrightarrow q$ are _____ and _____ respectively.
A. F, F
B. T, F
C. F, T
D.T, T
Last updated date: 22nd Mar 2023
•
Total views: 305.1k
•
Views today: 6.83k
Answer
305.1k+ views
Hint: Consider the two statements p and q. Prove that they both are true. if p and q are true then $p\to q$ also becomes true. If both are true $p\leftrightarrow q$ also becomes true i.e. p implies q and q implies p.
Complete step-by-step answer:
p: Every square is a rectangle.
(i) A rectangle is a quadrilateral with all 4 angles $90{}^\circ $.
From this definition, you can prove that the opposite sides are parallel and of the same lengths. A rectangle can be tall & thin, short & fat or all the sides can have the same length.
(ii) A square is a quadrilateral with all 4 angles right angles and all 4 sides of same length.
So a square is a special kind of rectangle, it is one where all the sides have the same length. Thus every square is a rectangle because it is a quadrilateral with all 4 angles right angles. However not every rectangle is a square, to be a square its sides must have the same length.
$\therefore $ p: Every square is a rectangle is true.
q: Every rhombus is a kite.
When all the angles are also $90{}^\circ $ the kite will be square. The sum of its side, opposite angles are equal. Therefore the rhombus is symmetrical about each of its diagonals. So a rhombus can become a kite. The difference between a rhombus and kite is that a kite doesn’t always have 4 equal sides or 2 pairs of parallel sides like rhombus.
$\therefore $ Every rhombus is a kite, but every kite is not a rhombus.
$\therefore $ q is true.
So we found out that both p and q are true.
$p\to q$ is T, it means that p implies q.
Also we know that $T\leftrightarrow T$ is true.
$\therefore $ $p\leftrightarrow q$ is true, it means that p implies q and q implies p.
Hence the truth values of $p\to q$ and $p\leftrightarrow q$ is T, T.
Note: If it was given,
p: Every rectangle is square.
q: Every kite is rhombus.
Then $p\to q$ would have been F.
As both the statements are wrong.
So, $p\leftrightarrow q$ is false.
So, the value of $p\to q$ and $p\leftrightarrow q$ becomes F, F or from the table of logical implication here p and q are T.
$\therefore $ $p\to q$ is T and $q\to p$is T i.e. case 1 is true.
Complete step-by-step answer:
p: Every square is a rectangle.
(i) A rectangle is a quadrilateral with all 4 angles $90{}^\circ $.
From this definition, you can prove that the opposite sides are parallel and of the same lengths. A rectangle can be tall & thin, short & fat or all the sides can have the same length.
(ii) A square is a quadrilateral with all 4 angles right angles and all 4 sides of same length.
So a square is a special kind of rectangle, it is one where all the sides have the same length. Thus every square is a rectangle because it is a quadrilateral with all 4 angles right angles. However not every rectangle is a square, to be a square its sides must have the same length.
$\therefore $ p: Every square is a rectangle is true.
q: Every rhombus is a kite.
When all the angles are also $90{}^\circ $ the kite will be square. The sum of its side, opposite angles are equal. Therefore the rhombus is symmetrical about each of its diagonals. So a rhombus can become a kite. The difference between a rhombus and kite is that a kite doesn’t always have 4 equal sides or 2 pairs of parallel sides like rhombus.
$\therefore $ Every rhombus is a kite, but every kite is not a rhombus.
$\therefore $ q is true.
So we found out that both p and q are true.
$p\to q$ is T, it means that p implies q.
Also we know that $T\leftrightarrow T$ is true.
$\therefore $ $p\leftrightarrow q$ is true, it means that p implies q and q implies p.
Hence the truth values of $p\to q$ and $p\leftrightarrow q$ is T, T.
p | q | $p\to q$ | $q\to p$ |
T | T | T | T |
T | F | F | T |
F | T | T | F |
F | F | T | T |
Note: If it was given,
p: Every rectangle is square.
q: Every kite is rhombus.
Then $p\to q$ would have been F.
As both the statements are wrong.
So, $p\leftrightarrow q$ is false.
So, the value of $p\to q$ and $p\leftrightarrow q$ becomes F, F or from the table of logical implication here p and q are T.
$\therefore $ $p\to q$ is T and $q\to p$is T i.e. case 1 is true.
Recently Updated Pages
Calculate the entropy change involved in the conversion class 11 chemistry JEE_Main

The law formulated by Dr Nernst is A First law of thermodynamics class 11 chemistry JEE_Main

For the reaction at rm0rm0rmC and normal pressure A class 11 chemistry JEE_Main

An engine operating between rm15rm0rm0rmCand rm2rm5rm0rmC class 11 chemistry JEE_Main

For the reaction rm2Clg to rmCrmlrm2rmg the signs of class 11 chemistry JEE_Main

The enthalpy change for the transition of liquid water class 11 chemistry JEE_Main

Trending doubts
Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Write a letter to the Principal of your school to plead class 10 english CBSE
