Logic is widely used not only in life, but also in the implementation of digital technology, including computers. Digital technology contains so-called logical elements that implement certain logical operations.

Logic uses simple and compound logical statements (narrative statements) that can be true ( 1 ) or false ( 0 ).

Example of simple statements:

• "Moscow is the capital of Russia" (1)
• "Twice two is three" (0)
• "Great!" (not a statement)

To combine several simple statements into one compound, logical operations are used. There are three basic logical operations: AND, OR, NOT.

Order of operations:

1. actions in brackets, comparison operations (<, ≤, >, ≥, =, ≠)

Let's consider each of the three operations separately.

1. Operation NOT changes the meaning of a logical statement to the opposite. This operation is also called "inversion", "logical negation". Operation sign: ¬

Truth table:

2. Operation I for a compound statement gives truth only if all the constituent simple statements are true. This operation can also be called "logical multiplication" or "conjunction". Operation sign: , & , /\

Truth table:

3. The OR operation for a compound statement gives truth when at least one of the input simple statements is true. "Logical addition", "disjunction". Operation sign: + , v

Examples of problem solving

Example 1.

For which of the following numbers is the statement false:

NOT(number > 50) OR(even number)?
1) 9 2) 56 3) 123 4) 8

Solution. First we perform comparisons in parentheses, then the NOT operation, and lastly the OR operation.

1) Substitute the number 9 into the expression:
NOT (9 > 50) OR(9 even)
NOT(lie) OR(false) = true OR false = true

9 does not suit us, since according to the condition we must receive a lie.

2) Substitute the number 56 into the expression:
NOT (56 > 50) OR(56 even)
NOT(true) OR(true) = false OR true = true

56 doesn't work either.

3) Substitute 123:
NOT (123 > 50) OR(123 even)
NOT(true) OR(false) = false OR false = false

The number 123 came up.

This problem could be solved in another way:
NOT(number > 50) OR(even number)

We need to get a false value. We see that the OR operation will be performed last. The OR operation will yield false when both the expressions NOT(number) and (even number) are false.

Since the condition (the number is even) must be equal to a false value, we immediately reject the options with the numbers 56, 8.

So, you can solve by direct substitution, which takes a long time and can cause an error when calculating the expression; or you can solve the problem quickly by analyzing all the simple conditions.

Answer: 3)

Example 2

For which of the given numbers is the following statement true:

NOT(First digit is even) AND NOT(Last digit is odd)?

1) 6843 2) 4562 3) 3561 4) 1234

First, we perform comparisons in parentheses, then NOT operations on parentheses, and lastly, the AND operation. This entire expression must evaluate to true.

Since the operation does NOT reverse the meaning of the statement, we can rewrite this complex expression as follows:

(First digit is odd) AND(Last digit is even) = true

As you know, logical multiplication AND gives truth only when all simple statements are true. So both conditions must be true:

(First digit is odd) = true (Last digit is even) = true

As you can see, only the number 1234 is suitable

Answer: 4)

Example 3

For which of the given names is the statement true:
NOT(First letter is vowel) AND(Number of letters > 5)?

1) Ivan 2) Nikolai 3) Semyon 4) Illarion

Let's rewrite the expression:
(First letter is not a vowel)AND(Number of letters > 5) = true
(First letter is consonant)AND(Number of letters > 5) = true

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