5 Express each of these statements using quantifiers Then fo

5. Express each of these statements using quantifiers. Then form the negation of the statement so that no negation is to the left of a quantifier. Next, express the negation in simple English. (Do not simply use the phrase \'\'It is not the case that.\'\') (a) Some student has solved every exercise in this book. (b) No one has climbed every mountain in the Himalayas. 6. Use quantifiers to express the distributive laws of multiplication over addition for real numbers.

Solution

The symbol ?means for all

The symbol ? means there exists or for some

These symbols are called quantifiers in truth tables or statements.

a) Some student has solved every exercise in this book.

Using quantifier:

? some student who has soutions ? exercise in this book.

--------------------------------------------------------------------------------------------------------

b) No one has climbed every mountain in the Himalayas.

Using quantifer:

? no one who is a climber ?mountains in the Himalayas.

------------------------------------------------------------------------------------------------------------------------

Negation of the given statements:

a) There is no student who has solved every exercise in this book.

b) Some one has climbed every mountain in the Himalayas.

----------------------------------------------------------------------------------------------------------------------------

6) Distributive law of multiplication over addition:

a(b+c) = ab+ac for all real numbers a, b and c.

?real numbers a,b and c, a(b+c) = ab+ac

.