Discrete Mathematical Structures Fundamentals of Logic Logical implication - Necessary and Sufficient conditions

1. By,
Lakshmi R
Asst. Professor
Dept. of ISE

2. Consider p →q.That is, if p, then q
then, p is sufficient condition for q and q is necessary condition for p
Eg: “If there are animals, then oxygen is present”
Presence of oxygen is necessary for animals’ life
Presence of animals is sufficient condition to say oxygen is present.
Oxygen can be present without the presence of animals.
Lakshmi R, Asst. Professor, Dept. of ISE

3. When a hypothetical p →q is such that q is true whenever p is true,
then we say that “p logically implies q”
This is symbolically represented as p ⇒q
‘⇒’ denotes logical implication
Consider two propositions p and q whose truth values are interrelated.
Then, for p →q to be a logical implication, following should hold good
i. p ⇒q
ii. p is sufficient for q
iii. q is necessary for p
Lakshmi R, Asst. Professor, Dept. of ISE

4. 1. If a number is divisible by 6, then it is
divisible by 3.
2. If a shape is a parallelogram, then its
diagonals bisect each other.
3. If a triangle is not isosceles, then it is not
equilateral.
4. If a quadrilateral is a square, then it is a
rectangle.
Lakshmi R, Asst. Professor, Dept. of ISE

5. If a number is divisible by 6, then it is divisible by
3.
A necessary condition for a number to be divisible by 6
is that it is divisible by 3.
A sufficient condition for a number to be divisible by 3 is
that number is divisible by 6.
Solution:
Lakshmi R, Asst. Professor, Dept. of ISE

6. If a shape is a parallelogram, then its diagonals bisect each
other.
A necessary condition for a shape to be parallelogram is that its diagonals
bisect each other.
A sufficient condition for diagonals of a shape to bisect each other is that it is a
parallelogram.
Solution:
Lakshmi R, Asst. Professor, Dept. of ISE

7. Lakshmi R, Asst. Professor, Dept. of ISE
If a triangle is not isosceles, then it is not equilateral.
Solution:
Consider contrapositive of the above statement.
If a triangle is equilateral, then it is isosceles
A sufficient condition triangle to be isosceles is that it is equilateral.
Note: triangle being isosceles does not guarantee that it is equilateral.
A necessary condition for a triangle to be equilateral is that it is isosceles.

8. Lakshmi R, Asst. Professor, Dept. of ISE
If a quadrilateral is a square, then it is a rectangle.
Shape Rules
Quadrilateral 4 sides
Parallelogram 4 sides
Opposite sides are parallel
Rectangle 4 sides
Opposite sides are parallel
Equiangular
Square 4 sides
Opposite sides are parallel
Equiangular
All sides are equal
Solve it!!