# Understanding the Basics of the De Morgan’s Laws

In the previous two posts, we have discussed about the logical operators and, or, and not.  Those two articles are preparation for this post.

Consider the following inequalities.

1.) $x \leq 8$

2.) $-2 < x < 3$

The first inequality can also be represented as $x < 8$ or $x = 8$

Now, how do we find

not ($x < 8$ or $x = 8$).

To easily understand the question, we graph $x \leq 8$ first and then see what’s not on that graph.  Continue reading

# Introduction to Negation and the NOT Operator

In the previous post, we have learned the difference between and and or. And and or are called logical operators. Statements that are connected by logical operators are called logical expressions.

In this post, we are going to learn another logical operator which is a logical operator for negation.

The negation of a given statement p is called not p, that if true, exactly expresses what it would mean to be false.

The truth table of the NOT operator is shown below. As we can see, a true statement when negated is true, and a false statement when negated is true.

Note that the negation of a statement is simply putting the not operator in front of it and not finding the opposite word for it. For example, the negation of beautiful is not beautiful; we cannot negate beautiful with ugly.

If two logical expressions have the same truth values, then, they are said to be logically equivalent.  For example, the expression p and not (not p) are logically equivalent as shown in the previous truth table.

# The difference between AND and OR

We use the word and and or many times in every day language. Consider the following statements.

I will cook lunch and I will clean the room.

I will cook lunch or I will clean the room.

In the first statement, the speaker implies that he will do both things. He will cook lunch and clean the room. Of course, it doesn’t matter if he will do these things separately or which will he do first. The word and in this statement means that he will do both chores.

In the second sentence, the speaker implies that he is choosing between cooking lunch and cleaning the room. Although he may do the other thing later, but at that particular moment, the word or in this statement means that he is trying to choose to do one of the two chores.  Continue reading

# Why the Multiplication by 11 Trick Works

One of the famous tricks that you have probably known is how to multiply numbers by 11. If we multiply a 2-digit number by 11, we copy the ones digit to the ones digit of the product, add the ones and the tens digit and place the sum in the tens digit of the product, and then copy the tens digit to the hundreds digit of the product. For example, if we multiply 34 × 11.

(1) We copy 4 and place it in the ones digit of the product.
(2) Add 3 and 4 = 7 and place it in the tens digit of the product.
(3) Copy 3 and place it in the hundreds digit of the product.

Therefore, 34 × 11 = 374.  Continue reading