#### Answer

1. Use truth tables to verify these equivalances.
a) p $\land$ T $\equiv$ p
p
0 | 0
1 | 1
b) p $\vee$ F $\equiv$ p
p
0 | 0
1 | 1
c) p $\land$ F $\equiv$ F
p
0 | 0
1 | 0
d) p $\vee$ T $\equiv$ T
p
0 | 1
1 | 1
e) p $\vee$ p $\equiv$ p
p
0 | 0
1 | 1
f) p $\land$ p $\equiv$ p
p
0 | 0
1 | 1

#### Work Step by Step

There are two propositional operators used here. $\land$(AND) and $\vee$(OR)
$\land$ takes two propositions, and evaluates to true if both propositions are true, so
false $\land$ false = false
false $\land$ true = false
true $\land$ false = false
true $\land$ true = true
Meanwhile, $\vee$ takes in two propositions, and evaluates to true if either or both propositions are true, so
false $\vee$ false = false
false $\vee$ true = true
true $\vee$ false = true
true $\vee$ true = true
We can substitute 0 to denote (F)alse, and 1 to denote (T)rue when writing our work. A truth table simply shows all possible values for the propositions in an expression, and tells us the corresponding truth value of the expression with those inputs.
In this case, since all of these expressions only contain 1 unique proposition, all of the truth tables will only contain 2 rows, for when p is 0(false), and 1(true), and 2 columns(the column for p, and the resultant column)
Take for instance part
a) p $\land$ T $\equiv$ p
if p = 0, then
p $\land$ 1 = 0 $\land$ 1 = 0
if p = 1, then
p $\land$ 1 = 1 $\land$ 1 = 1
These two observations are translated into the truth table
p
0 | 0
1 | 1
And since the column of the truth table with p, is equivalent to the resultant column of the expression,
the result of the expression is equivalent to the truth value of p, which is expressed in the statement
p $\land$ T $\equiv$ p
We shall prove the next equivalencies similarly
b) p $\vee$ F $\equiv$ p
if p = 0
p $\vee$ 1 = 0 $\vee$ 1 = 0
if p = 1
p $\vee$ 1 = 1 $\vee$ 1 = 1
p
0 | 0
1 | 1
c) p $\land$ F $\equiv$ F
if p = 0
p $\land$ 0 = 0 $\land$ 0 = 0
if p = 1
p $\land$ 0 = 1 $\land$ 0 = 0
p
0 | 0
1 | 0
d) p $\vee$ T $\equiv$ T
if p = 0
p $\vee$ 1 = 0 $\vee$ 1 = 1
if p = 1
p $\vee$ 1 = 1 $\vee$ 1 = 1
p
0 | 1
1 | 1
e) p $\vee$ p $\equiv$ p
if p = 0
p $\vee$ p = 0 $\vee$ 0 = 0
if p = 1
p $\vee$ p = 1 $\vee$ 1 = 1
p
0 | 0
1 | 1
f) p $\land$ p $\equiv$ p
if p = 0
p $\land$ p = 0 $\land$ 0 = 0
if p = 1
p $\land$ p = 1 $\land$ 1 = 1
p
0 | 0
1 | 1