Truth table of p -> q Easy

The question

Question: make the truth table of p → q.

1 Rows for p and q

Each letter can be true or false.

Two letters give 2 × 2 = 4 rows: every way they can go.

We make a table: a column for each letter, then a column for each part, from the inside out.

We write T for true and F for false.

p: the first half T, the second half F.

This way no row is missed.

q: T, F, T, F, down the column.

2 Column for p → q

We read the arrow as if, then.

p → q is a promise: if p, then q.

It is broken only when p is true and q is false.

So it is F only in row 2.

Careful!

If the if-part is false, the promise is not broken.

So F → F is T, and so is F → T.

The last column is the answer: when p → q is true.

Top to bottom: T, F, T, T.

3 Check

Let's check row 2 by hand.

In row 2, p is true and q is false.

Put them in: T → F.

It gives F, as in the table.

Right!

Let's look back.

First: rows for p and q.

Then: column for p → q.

Last: check.

▶ Watch it solved

Truth table of ~(p or q) Medium

The question

Question: make the truth table of ¬(p ∨ q).

1 Rows for p and q

Each letter can be true or false.

Two letters give 2 × 2 = 4 rows: every way they can go.

We make a table: a column for each letter, then a column for each part, from the inside out.

We write T for true and F for false.

p: the first half T, the second half F.

This way no row is missed.

q: T, F, T, F, down the column.

2 Column for p ∨ q

We read this sign as or.

p ∨ q is true when at least one of them is true.

It is false only when both are false.

So it is F only in row 4.

3 Column for ¬(p ∨ q)

We read this sign as not.

¬(p ∨ q) is the opposite of p ∨ q: T becomes F, F becomes T.

So it is T only in row 4.

Careful!

The ¬ is for the whole bracket, not only the first letter.

The last column is the answer: when ¬(p ∨ q) is true.

Top to bottom: F, F, F, T.

4 Check

Let's check row 1 by hand.

In row 1, p is true and q is true.

Put them in: ¬(T ∨ T).

The inside first.

It gives F, as in the table.

Right!

Let's look back.

First: rows for p and q.

Then: column for p ∨ q.

Then: column for ¬(p ∨ q).

Last: check.

▶ Watch it solved

Truth table of (p and q) -> r Harder

The question

Question: make the truth table of (p ∧ q) → r.

1 Rows for p, q and r

Each letter can be true or false.

Three letters give 2 × 2 × 2 = 8 rows: every way they can go.

We make a table: a column for each letter, then a column for each part, from the inside out.

We write T for true and F for false.

p: the first half T, the second half F.

This way no row is missed.

q: two T, two F, and again.

r: T, F, T, F, down the column.

2 Column for p ∧ q

We read this sign as and.

p ∧ q is true only when p and q are both true.

So it is T only in rows 1 and 2.

3 Column for (p ∧ q) → r

We read the arrow as if, then.

(p ∧ q) → r is a promise: if p ∧ q, then r.

It is broken only when p ∧ q is true and r is false.

So it is F only in row 2.

Careful!

If the if-part is false, the promise is not broken.

So F → F is T, and so is F → T.

The last column is the answer: when (p ∧ q) → r is true.

Top to bottom: T, F, T, T, T, T, T, T.

4 Check

Let's check row 2 by hand.

In row 2, p is true, q is true and r is false.

Put them in: (T ∧ T) → F.

The inside first.

It gives F, as in the table.

Right!

Let's look back.

First: rows for p, q and r.

Then: column for p ∧ q.

Then: column for (p ∧ q) → r.

Last: check.

▶ Watch it solved