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.)

2.)

The first inequality can also be represented as **or** .

Now, how do we find

**not** ( **or** ).

To easily understand the question, we graph first and then see what’s not on that graph.

As shown above, the points not on the graph of are points on the graph of . They are points that are not on and not on or points on

**not** **and not** .

Since the two logical statements are equivalent, we can write

**not** ( **or** ) not ()** and not** ().

Now, if we represent as p and as q we can have the following equivalence.

**not (p or q)** ** (not p) and (not q)*****.**

Let us consider the second inequality.

The inequality can also be represented as **and** .

Now, how do we find

**not** ( **and** )?

Again, we are searching for the part of the graph that is not on the graph of . As shown below, it should be or .

Therefore,

**not** (x > -2 and x < 3) **not** ()$ **or** **not**

If we represent this expression by or by , we can say that

**not (p and q) ** **(not p) or (not q) ****

The relationships in * and ** are called the De Morgan’s Laws named after British mathematician and logician** Augustus De Morgan**.

We can prove the De Morgan’s Laws by examining the truth tables of the identities above. First, we see the truth tables of **not (p and q)** and **not (p or q)**.

Then, we also examine the truth table of** not p or not q** and **not p and not q**.

As we can see, the identity **not (p and q)** in the first table is the same as **not p or not q** in the second table. Also, the truth values of **not (p or q)** in the first table is the same as **not p and not q** in the second table. This proves the conjecture in * and ** are logically equivalent. This is one of the proofs of the De Morgan’s Laws.