A relation R is given in the matrix below. Rows of the matrix are numbered 1 through 4 from top to bottom and columns are numbered 1 through 4 from left to right. Select the expression that is false.

Question

BlackTom AI · Accepted Answer

Matrix entry (i,j) = 1 means iRj holds. The given 4×4 matrix (rows 1–4, columns 1–4) is:
Row1: [0, 1, 0, 0]
Row2: [0, 1, 0, 1]
Row3: [0, 1, 0, 1]
Row4: [1, 0, 0, 0]

1) Relevant knowledge points and concepts
- A relation R on a finite set can be represented by a 0–1 matrix: entry a_ij = 1 ⇔ iRj is true.
- To test a statement like iRj, read the matrix at row i, column j.

2) Complete reasoning process step by step
- Map each candidate expression to the corresponding matrix entry:
  - 2R2 corresponds to entry (row2, col2).
  - 3R4 corresponds to entry (row3, col4).
  - 4R1 corresponds to entry (row4, col1).
  - 4R3 corresponds to entry (row4, col3).

3) Calculations / intermediate steps (reading entries)
- Entry (2,2) = Row2 second element = 1 → 2R2 is true.
- Entry (3,4) = Row3 fourth element = 1 → 3R4 is true.
- Entry (4,1) = Row4 first element = 1 → 4R1 is true.
- Entry (4,3) = Row4 third element = 0 → 4R3 is false.

4) Why the correct answer is correct
- The only candidate that corresponds to a 0 in the matrix is 4R3 (row 4, column 3 = 0). Since a 0 means the ordered pair (4,3) is not in R, the statement 4R3 is false.

5) Why the other options are incorrect (i.e., why they are true)
- 2R2 is incorrect as a choice for "false" because entry (2,2)=1, so 2R2 is true.
- 3R4 is true because entry (3,4)=1.
- 4R1 is true because entry (4,1)=1.
Thus none of these are false.

6) Conclusion
The only false expression is 4R3, which matches the provided correct answer: 4R3.

A relation R is given in the matrix below. Rows of the matrix are numbered 1 through 4 from top to bottom and columns are numbered 1 through 4 from left to right. Select the expression that is false.Single choice

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